^dm.. 


1 


MV/R! 


MECHANICS 


PROBLEMS 


FOR  ENGINEERING  STUDENTS 


BY 

FRANK   B.   SANBORN 

Member  of  American  Society  of  Ciiil  Engineers 
Professor  cj  Civil  Engineering  in  Tujls  College 


SECOND  EDITION,  REVISED  AND  ENLARGED 
FIRST  THOUSAND 


UNIVERSITY 

OF 

&^^_P^^y^         NEW   YORK 

JOHN   WILEY   &    SONS 

London:    CHAPMAN   &   HALL,   Limited 

1906 


PENERAL 


Copyright,  1902, 1906, 

BY 

FRANK  BERRY  SANBORN 


PREFACE 


There  is  an  opinion  among  engineers  that  too 
often  students  are  not  well  grounded  in  the  practical 
problems  of  Mechanics ;  that  they  know  more  of 
theory  and  mathematical  deductions  than  of  practical 
applications.  A  prominent  educator  has  recently 
said  to  me,  in  regard  to  the  teaching  of  Mechanics, 
"  I  am  convinced  that  it  is  to  be  done  more  thoroughly 
in  the  future  than  in  the  past  ;  "  and  it  will  be  done, 
he  believes,  by  sticking  close  to  elementary  principles 
as  developed  by  well-chosen  practical  problems.  Fur- 
thermore, he  adds,  "  it  will  have  to  be  recognized  that 
all  an  engineering  baccalaureate  course  can  worthily 
accomplish  is  to  give  the  raw  recruit  the  '  setting-up ' 
exercises  in  Mechanics." 

It  is  now  generally  recognized,  I  think,  that  this 
subject  should  cover  first  of  all  the  elements  and 
fundamental  principles  that  form  the  basis  of  every 
engineer's  knowledge  ;  that  these  necessary  elements 
and  principles  are  best  understood  and  best  remem- 
bered by  actually  solving  numerous  problems  that 
present  important  facts  illustrative  of  every-day  engi- 
neering practice,  and  arouse  the  student's  interest 
far  better  than  abstract  examples  which  can  be  easily 
formulated  from  imaginary  conditions. 

Therefore,  for  the  reasons  indicated  above,  an  effort 


149159 


IV  PREFACE. 

has  been  made  in  preparing  this  book  to  present,  from 
actual  conditions,  many  practical  problems  together 
with  brief  definitions  and  solutions  of  typical  prob- 
lems which  should  help  the  student  in  Mechanics 
to  follow  the  advice  once  given  by  George  Stephenson 
to  his  son  Robert  : 

"  Learn  for  yourself,  think  for  yourself, 
make  yourself  master  of  principles." 

Photographs  or  electroplates  have  been  furnished 
for  certain  of  the  illustrations  as  follows : 

Page  17  by  Otto  Gas  Engine  Works;  pages  20  and  32,  Pelton 
Water  Wheel  Company  ;  page  24,  Wellington- Wild  Coal  Company  ; 
page  25,  Harrisburg  Foundry  and  Machine  Company;  page  29, 
Fall  River  Iron  Works  Company ;  page  35,  Associated  Factory 
Mutual  Fire  Insurance  Companies;  page  d^,,  Maryland  Steel  Com- 
pany; page  64,  Bucyrus  Company;  page  120,  A.  J.  Lloyd  &  Co.; 
page  146,  CUnton  Wire  Cloth  Company;  page  148,  The  Detroit 
Graphite  Manufacturing  Company ;  page  149,  The  Engineering 
Record;  pages  151  and  169,  Brown  Hoisting  and  Conveying  Machin- 
ery Company;  page  153,  Cement  Age;  page  157,  Fig.  84,  American 
Locomotive  Company ;  page  161,  Carson  Trench  Machinery  Com- 
pany ;  page  164,  Chicago  Bridge  and  Iron  Works  ;  page  165,  Chap- 
man Valve  Manufacturing  Company. 

FRANK   B.    SANBORN. 

Tufts  College,  Mass., 
June,  1906. 


CONTENTS 


I.    WORK. 

Problems  i  to  172. 
FOOT-POUNDS  PAGE 

Raising  weights,  overcoming  resistances  of  railroad 
trains,  macliine  punch,  construction  of  wells  and 
chimneys,  operation  of  pumping  engines.  Force  and 
distance  or  foot-pounds  required  in  cases  of  pile- 
driver,  horse,  differential  pulley,  tackle,  tram  car      .     .      7 

HORSE-POWER 

Required  by  windmills,  planing  machines,  gas  engine, 
locomotive,  steam  engines  —  simple,  compound,  triple, 
slow  speed,  high  speed  engines.  Horse-power  from 
indicator  cards,  required  by  electric  lamps,  driving- 
belts,  steam  crane,  coal  towers,  pumping  engine, 
canals,  streams,  turbines,  water-wheels.  Efficiency, 
force  or  distance  required  in  cases  of  fire  pumps, 
mines,  bicycles,  shafts,  railroad  trains,  air  brakes,  the 
tide,  electric  motors,  freight  cars,  ships 16 

ENERGY 

Foot-pounds,  horse-power,  velocity:  —  Ram,  hoisting- 
engine,  blacksmith,  electric  car,  bullet,  cannon,  nail, 
pendulum.  Energy  resulting  from  motion  of  fly-wheel 
and  energy  required  by  jack-screw 44 


vi  CONTENTS. 

II.    FORCE. 

Problems  172  to  414. 

FORCES  ACTING   AT   A    POINT  pagb 

Canal  boat  being  towed,  rods,  struts,  beams,  derrick, 
cranes  set  as  in  action ;  balloon  held  by  rope,  ham- 
mock supported  ;  wagon,  trucks,  picture  supported  ; 
forces  in  frames  of  car  dumper,  tripod,  shear  legs, 
dipper  dredge  ;  also  in  triangle,  square,  sailing  vessel, 
rudder,  foot-bridge,  roof-truss ....    51 

MOMENTS    FOR    PARALLEL   FORCES 

Beam  balanced,  pressure  on  supports,  propelling  force 
of  oars,  raising  anchor  force  at  capstan,  bridge  loaded 
pressure  on  abutments,  lifting  one  end  of  shaft,  boat 
hoisted  on  davit,  forces  acting  on  triangle,  square, 
supports  of  loaded  table  and  floor        72 

COUPLES 

Brake  wheel,  forces  acting  on  square 84 

STRESSES 

Beam  leaning  against  wall,  post  in  truss,  rope  pull  on 
chimney,  connecting  rod  of  engines,  trap-door  held  up 
by  chain 86 

CENTER   OF   GRAVITY 

Rods  with  loads,  metal  square  and  triangle,  circular 
disk  with  circular  hole  punched  out,  box  with  cover 
open,  rectangular  plane  with  weight  on  one  end, 
irregular  shapes,  solid  cylinder  in  hollow  cylinder, 
cone  on  top  of  hemisphere 90 

FRICTION 

Weight  moved  on  level  table,  stone  on  ground, 
block  on  inclined  plane,  gun  dragged  up  hill,  cone 
sliding  on  inclined  plane  ;  friction  of  planing  machine. 


CONTENTS.  vii 

PAGE 

locomotives,  trains,  ladder  against  wall,  bolt  thread, 
rope  around  a  post ;  belts,  pulleys  and  water-wheels 
in  action  ;  heat  generated  in  axles  and  bearings.      .     .    96 


III.    MOTION. 

Problems  414  to  527. 

UNIFORM    ACCELERATION 

Railroad  train,  ice  boat,  stone  falling  and  depth  of 
well,  balloon  ascending,  cable  car  running  wild.   .     .     .119 

RELATIVE  VELOCITY 

Aim  in  front  of  deer,  rowing  across  river,  bullet  hit- 
ting balloon  ascending,  rain  on  passenger  train,  wind 
on  steamer,  two  passing  railroad  trains 126 

DISTANCE,  VELOCITY,  FRICTION,  ANGLE  OF 
INCLINATION 
Train  stopped,  steamer  approaching  dock,  cannon 
recoil,  locomotive  increasing  speed,  body  moved  on 
table,  box-machine,  motion  of  table,  barrel  of  flour  on 
elevator,  man's  weight  on  elevator,  cage  drawn  up 
coal  shaft.         121 

PROJECTILES 

Inclination  for  bullet  to  strike  given  point,  motion 
down  plane,  stone  dropped  from  train,  thrown  from 
tower,  projectile  from  hill,  from  bay  over  fortification 
wall 133 

PENDULUMS 

Simple,  conical,  ball  in  passenger  car 141 

IMPACT 

Water  suddenly  shut  off,  cricket  ball  struck,  hammer 
falling  on  pile,  shot  from  gun,  bullet  from  rifle,  freight 
and  passenger  trains  collide 142 


Vlll  CONTENTS, 

REVIEW. 
Problems  528  to  600. 

PRACTICAL  PROBLEMS 

Water  turbine  test,  suspension  bridge,  Niagara  tower, 
launching  data,  coal-wharf  incline,  typical  American 
bridge,  modern  locomotive  tests,  wood  in  compression, 
actual  cableway,  St.  Elmo  water-tower,  outside-screw- 
and-yoke  valve,  cast-iron  pipe,  retaining  walls,  geared 
drum,  gas-engine  test 145 

EXAMINATIONS 

Yale,  Tufts,  Harvard 174 

ANSWERS 

600  problems,  besides  43  under  Examinations.  About 
one-half  have  answers  given 1S4 

DEFINITIONS 

Work,  force,  and  motion  and  their  sub-divisions    ....       2 

TABLES 

FalHng  Bodies,  Functions  of  Angles,  Unit  Values  — 
heights  and  velocities •  190 

INDEX 193 


MECHANICS-PROBLEMS. 

INTRODUCTION. 

The  problems  and  solutions  that  follow  have  been 
arranged  in  the  order  of  Work,  Force,  and  Motion. 
At  the  beginning  of  each  important  section  one 
problem  has  been  solved  so  as  to  explain  the  method 
of  solving  similar  problems  and  to  serve  as  a  guide  for 
solutions  to  be  put  in  note-books.  An  effort  has  been 
made  throughout  the  book  to  simplify.  Few  methods 
have  been  presented  ;  the  calculus  has  been  used  only 
where  necessary  ;  no  discussion  has  been  offered  of 
the  term  mass  —  many  such  subjects  have  been  left 
for  more  advanced  courses  or  extended  treatises. 

The  "  gravitation  system  "  of  units  —  the  foot- 
pound-second system,  or  meter-kilogram-second  sys- 
tem—  known  as  the  engineers'  system  has /been 
used  exclusively. 

In  engineering  practice  one  is  often  puzzled  to  tell 
just  what  data  to  collect  and  afterward  how  much  of 
it  to  use  ;  because  of  this,  I  have  left  more  data  in 
some  of  the  problems,  and  especially  those  under 
Review,  than  is  absolutely  necessary  for  solving  the 
problem,  and  the  student  will  have  opportunity  '*  to 
pick  and  choose  "  just  as  he  would  do  in  actual  cases. 


ME  CIIA  NICS  -  PR  OBLEMS. 


DEFINITIONS. 

Mechanics  is  the  science  that  treats  of  the  action 
of  forces  at  rest  and  in  motion. 

Work,  Force,  and  Motion  are  the  three  sub-divisions 
of  Mechanics  considered  in  this  book. 

WORK. 

Work  is  done  by  the  action  of  force  through  some 
distance. 

Work  is  measured  by  the  product  of  force  times  the 
distance  through  which  it  acts. 

Work  =  force  x  distance,  —  a  formula  fundamental 
for  all  Work  problems. 

Energy  is  the  amount  of  work  that  a  body  possesses. 

Potential  energy  is  the  work  that  a  body  possesses 
by  virtue  of  its  position  above  the  earth's  surface. 

Kinetic  energy  is  the  work  that  a  body  possesses  by 
virtue  of  its  velocity. 

Horse-power  is  the  rate  of  doing  work.  One  horse- 
power is  the  equivalent  of  33  000  foot-pounds  of 
work  done  per  minute. 

FORCE. 

Force  in  Mechanics  has  both  magnitude  and  direc- 
tion, and  in  this  treatise 


DEFINITIONS.  3 

Force  Magnitude  is  usually  expressed  in  pounds. 
It  may  act  as  pressure,  a  push,  or  as  tension,  a  pull. 

Concurrent  forces  acting  on  a  body  arc  those  that 
have  the  same  point  of  application. 

Non-concurrent  have  different  points  of  application. 

Moment  of  a  force  about  a  point  or  axis  is  the 
product  obtained  by  multiplying  the  magnitude  of 
the  force  by  the  shortest  distance  from  the  point  or 
axis  to  the  line  of  action  of  the  force. 

Moment  =  force  x  perpendicular.  Clockwise  ten- 
dency of  rotation  is  usually  taken  positive. 

Resultant  of  a  system  of  concurrent  forces  is  a 
single  force  that  might  be  substituted  for  them  with- 
out changing  the  effect. 

Equilibriant  of  a  system  of  forces  is  a  single  force 
that  balances  them.  The  equilibriant  is  equal  and 
opposite  to  the  resultant. 

Components  of  a  single  force  are  the  forces  that 
might  be  substituted  for  it  without  changing  the 
effect. 

Parallelogram  of  forces.  When  three  forces  that 
are  in  equilibrium  meet  in  a  point  they  can  be  repre- 
sented in  magnitude  and  direction  by  the  sides  of  a 
parallelogram.  This  parallelogram  is  called  the 
parallelogram  of  forces. 


4  MECHANICS-PROBLEMS. 

1.  S  Vertical  components  =  o.  When  the  forces 
acting  in  one  plane  upon  a  body  are  in  equihbrium, 
the  forces  can  be  resolved  into  components  in  any 
one  direction,  and  the  algebraic  sum  of  the  compo- 
nents will  equal  o.     Likewise, 

2.  2  Horizontal  components  =  o.  The  algebraic 
sum  of  the  components  in  a  direction  perpendicular 
to  that  of  I  will  equal  zero  ;  and 

3.  S  Moments  =  o.  The  algebraic  sum  of  the 
moments  of  the  forces  taken  about  any  point  or  axis 
in  the  plane  will  equal  zero. 

These  three  axioms  can  frequently  be  used  to 
formulate  three  equations  that  contain  unknown 
quantities  which  can  then  be  determined. 

A  Couple  consists  of  two  equal,  opposite,  parallel 
forces  not  acting  in  the  same  straight  line. 

Moment  of  a  couple  is  the  product  of  one  of  the 
equal  forces  by  the  perpendicular  distance  between 
them. 

Center  of  gravity  of  a  body  or  a  system  of  bodies  is  a 
point  about  which  the  body  or  system  can  be  imagined 
to  balance  and  the  forces  of  gravity  will  cause  no 
rotation. 

Centroid  and  Center  of  Mass  are  terms  that  are 
sometimes  used  in  preferenc?  to  center  of  gravity. 

Centroid  is  the  point  of  application  of  a  system  of 
parallel  forces. 


DEFINITIONS. 


MOTION. 


Motion  (uniform)  is  that  in  wliich  a  body  moves 
through  equal  distances  in  equal  times. 

Motion  (accelerated)  is  that  in  which  a  body  moves 
through  unequal  distances  in  equal  times. 

Motion  (uniform-accelerated)  is  that  in  which  the 
velocity  increases  the  same  amount  in  each  unit  of 
time,  which  is  generally  taken  as  the  second. 

Acceleration  is  the  gain  or  loss  in  velocity  per  unit 
of  time. 

Centrifugal  force.  When  a  body  is  compelled  to 
move  in  a  curved  path  it  exerts  a  force  directed  out- 
wards from  the  center ;  its  amount  is  the  centrifugal 

W  If" 
force  =  

g    r 

Impact  is  said  to  take  place  when  one  body  strikes 
against  another. 

A  period  of  compression  thus  occurs,  and  the  forces 
acting  are  Impulsive  forces  of  compression.  Then 
follows  a  period  of  restitution. 

Coefficient  of  restitution  e  for  any  pair  of  substances 
is  the  ratio  of  the  impulsive  force  of  restitution  to  the 
impulsive  force  of  compression. 


/■  >''   OK  THE 


I 


UNIVERSITY 

OF 


I.     WORK. FOOT-POUNDS. 

1.  A  20th-century  express  of  100  tons  weight  goes 
up  a  grade  of  i  vertical  in  1 20  horizontal  ;  the  re- 
sistances are  1 5  pounds  per  ton.  Find  the  amount 
of  work  that  locomotive  expends  per  mile  of  travel. 


Work 

of  locomotive 

=  work          +  work 

of  friction              of  lifting  train 

Work 

of  friction 

=  force  X  distance 

Force 

=  15  X  100 

=  I  500  pounds 

Distance 

=  I  mile 

=  5  280  feet 

Work 

of  friction 

=  I  500  X  5  280 

=  792  000  foot-pounds 
Work  =  force  X  distance 

of  lifting  train 

Force  =  100  x  2  000 

=  200  000  pounds 
Distance        =  5  2  8dNX  y^^ 
=  44  feet 
.-.  Work  =  200  000  X  44 

of  lifting  train 

=  8  800  000  foot-pound 
Work      =  792  000  +  8  800  000 

of  locomotive 

=  9  592  000  foot-pounds 

2.  Find  the  work  done  by  an  cni;ine  in  drawing  a 
train  i  mile  along  a  level  railroad,  w  hen  the  constant 
resistances  of  friction,  air,  and  so  on,  are  i  ton. 

7 


8  MECHANICS-PROBLEMS. 

3.  A  punch  exerts  a  uniform  pressure  of  36  tons 
in  punching  a  hole  through  an  iron  plate  of  one-half 
inch  thickness.      Find  the  foot-pounds  of  work  done. 

4.  Find  what  work  is  being  done  per  minute  by 
an  engine  that  is  raising  2  000  gallons  of  water  an 
hour  from  a  mine  300  feet  deep. 

5.  If  a  weight  of  i  130  povmds  be  lifted  up  20 
feet  by  20  men  twice  in  a  minute,  how  much  work 
does  each  man  do  per  hour  } 

6.  A  number  of  men  can  each  do,  on  the  average, 
495  000  foot-pounds  of  work  per  day  of  8  hours. 
How  many  such  men  are  required  to  do  33  000  x  10 
foot-pounds  of  work  per  minute.? 

7.  A  centrifugal  pump  delivers  water  10  feet 
above  the  level  of  a  lake  of  half  a  square  mile  area. 
At  the  end  of  a  day's  pumping,  the  water  has  been 
lowered  i|^  feet.      How  much  work  has  been  done  } 

"  Distance  "  will  be  lo  feet  plus  |  of  1 1  feet. 

8.  Water  in  a  well  is  20  feet  below  the  surface  of 
the  ground,  and  when  500  gallons  have  been  pumped 
out  it  is  26  feet  below.     Find  the  work  done. 

9.  Brick  and  mortar  for  a  chimney  100  feet  high 
are  raised  to  an  average  height  of  35  feet.  Total 
amount  of  material  used  40  000  cubic  feet  or  about 
5  600  000  pounds.     What  work  was  done  } 

10.  What  work  is  done  in  winding  up  a  chain  that 
hangs  vertically,  is  130  feet  long,  and  weighs  20 
pounds  per  foot  1 


IVOKJC  —  FO  O  r-PO  UNDS.  9 

11.  A  chain  of  weight  300  pounds  and  length  i  50 
feet,  with  a  weight  of  500  pounds  at  the  end  of  it,  is 
wound  up  by  a  capstan.     What  work  is  done  ? 

12.  A  stream  of  width  20  feet,  average  depth  3 
feet,  and  mean  velocity  of  3  miles  per  hour  has  an 
available  fall  of  80  feet.  What  work  is  stored  in  the 
quantity  of  water  flowing  each  minute  ? 

Find  tlie  pounds  of  water  flowing  by  observing  that 
Quantity  =  area  X  velocity. 

13.  A  horse  draws  150  pounds  of  earth  out  of  a 
well,  by  means  of  a  rope  going  over  a  fixed  pulley, 
which  moves  at  the  rate  of  2\  miles  an  hour.  Neg- 
lecting friction,  how  many  units  of  work  does  this 
horse  perform  a  minute  } 

14.  A  cylindrical  shaft  14  feet  in  diameter  must  be 
sunk  to  a  depth  of  10  fathoms  through  chalk,  the 
weight  of  which  is  144  pounds  per  cubic  foot.  Find 
the  work  done. 

15.  A  well  is  to  be  dug  20  feet  deep  and  4  feet  in 
diameter.  Find  the  work  in  raising  the  material,  sup- 
posing that  a  cubic  foot  of  it  weighs  140  pounds. 

16.  A  horse  draws  earth  from  a  trench  by  means 
of  a  rope  going  over  a  pulley.  He  pulls  up,  twice 
every  5  minutes,  a  man  weighing  130  pounds,  and  a 
barrowful  of  earth  weighing  260  pounds.  Each  time 
the  horse  goes  forward  40  feet.  Find  the  useful 
work  done  per  hour. 

17.  A  body  weighing  50  pounds  slides  a  distance 


lO  MECHANICS-PROBLEMS. 

of  8  feet  down  a  plane  inclined  20°  to  the  horizontal, 
against  a  constant  retarding  force  of  4  pounds. 
Compute  the  total  work  done  upon  the  body  by 
(gravity)  its  weight  and  the  friction. 

18.  What  work  is  stored  in  a  cross-bow  whose 
cord  has  been  pulled  1 5  inches  with  a  maximum  force 
of  224  pounds  } 

19.  If  25  cubic  feet  of  water  are  pumped  every 
5  minutes  from  a  mine  140  fathoms  deep,  what 
amount  of  work  is  expended  per  minute  ? 

20.  In  pumping  i  000  gallons  from  a  water-cistern 
with  vertical  sides  the  surface  of  the  water  is  lowered 
5  feet.  Find  the  work  done,  the  discharge  being  10 
feet  above  the  original  surface. 

21.  A  uniform  beam  weighs  i  000  pounds,  and  is 
20  feet  long,  it  hangs  by  one  end,  round  which  it  can 
turn  freely.  How  many  foot-pounds  of  work  must  be 
done  to  raise  it  from  its  lowest  to  its  highest 
position  ? 

22.  A  body  is  suspended  by  an  elastic  string  of 
unstretched  length  4  feet.  Under  a  pull  of  10 
pounds  the  string  stretches  to  a  length  of  5  feet. 
Required  the  work  done  on  the  body  by  the  tension 
of  the  string  while  its  length  changes  from  6  feet  to 
4  feet. 

23.  A  weight  of  200  pounds  is  to  be  raised  to  a 
height  of  40  feet  by  a  chord  passing  over  a  fixed 
smooth  pulley  ;  it  is  found  that  a  constant  force  P, 
pulling  the  chord  at  its  other  end  for  three-fourths  of 


WORK  —  FO  O  T-PO  UNDS.  1 1 

the  ascent,  communicates  sufficient  velocity  to  the 
weight  to  enable  it  to  reach  the  required  height. 
Find  P. 

Work        =  force  x  distance 

Work        =200     X  40 

on  weight 

Work        =  P  X  2  of  40 

by  pull 

Work        =  Work 

on  weight  by  pull 

200  X  40  =  P  X  30 

P  =  2665  pounds 

24.  A  horse  drawing  a  cart  along  a  level  road  at 
the  rate  of  2  miles  per  hour  performs  29  216  foot- 
pounds of  work  in  3  minutes.  What  pull  in  pounds 
does  the  horse  exert  in  drawing  the  cart  > 

25.  It  is  said  that  a  horse  can  do  about  1 3  200  000 
foot-pounds  of  work  in  a  day  of  8  hours,  walking  at 
the  rate  of  2^  miles  per  hour.  What  pull  in  pounds 
could  such  a  horse  exert  continuously  during  the 
working-day  } 

26.  If  a  horse  walking  once  round  a  circle  i  o  yards 
across  raises  a  ton  weight  18  inches,  what  force  does 
he  exert  over  and  above  that  necessary  to  overcome 
friction  1 

27.  A  building  of  weight  50  000  pounds  is  being 
moved  on  rollers  by  a  horse  that  is  pulling  on  a  pole 
with  a  distance  of  10  feet  from  the  center  of  a  capstan 
that  is  1 8  inches  in  diameter.  If  the  total  friction  is 
200  pounds  per  ton,  what  force  must  the  horse  exert  ? 


12 


ME  CHA  NICS-rROBLEMS. 


28.  The  500-pound  hammer  of  a  pile-driver  is 
raised  to  a  height  of  20  feet  and  then  allowed  to 
fall  upon  the  head  of  a  pile,  which  is  driven  into  the 
ground  i  inch  by  the  blow.  Find  the  average  force 
which  the  hammer  exerts  upon  the  head  of  the  pile. 

Work  =  force  X  distance 
=  500  X  20 
=  10  000  foot-pounds 
Distance  =tV  ^^^^ 
/.  10  000  foot-pounds  =  force  x  iV  ^^^t 
.-.  force  =10  000  X  12 

=  120  000  pounds 

29.  A  hammer  weighing  i  ton  falls  from  a  height 
of  24  feet  on  the  end  of  a  vertical  pile,  and  drives  it 
half  an  inch  deeper  into  the  ground.  Assume  the 
driving  force  of  the  hammer  on  the  pile  to  be  con- 
stant while  it  lasts,  and  find  its  amount  expressed  in 
tons  weight. 

30.    Determine  by  the  principle  of  work, 
neglecting  friction,  the  relation  between  the 
pull  P  and  the  load  W  in  case  of  the  differ- 
ential wheel-and-axle  of  Fig.  i. 
For  one  revolution, 

Work         =  P  X  2  TT^ 

of  P 

Work         =  I  W  X  2  -nr'  -  ^  W  X  2  irr 

on  weight 

P  X  2  TT^?  =  I  \\V  X  2  (r'  —  r) 
P  X  2  ^  =  W  {?■'  -  r) 
W  _  7'  -  r 
"P  ~      2a     * 


WORK—FO  O  T-PO  UNDS.  I  3 

31.  A  barrel  of  Portland  cement  that  weighs  396 
pounds  is  to  be  hoisted  by  a  wheel  and  axle  as  in  Fig. 
I  ;  the  radii  are  6,  12  and  18  inches.  What  force 
will  be  required  ? 

32.  If,  neglecting  frictions,  a  power  of  10  pounds, 
acting  on  an  arm  2  feet  long,  produces  in  a  screw- 
press  a  pressure  of  half  a  ton,  what  would  be  the 
pitch  of  the  screw  .? 

33.  What  is  the  ratio  of  the  weight  to  the  power, 
in  a  screw-press  working  without  friction,  when  the 
screw  makes  4  turns  in  the  inch,  and  the  arm  to 
which  the  power  is  applied  is  2  feet  long  ? 

34.  What  force  applied  at  the  end  of  an  arm 
18  inches  long  will  produce  a  pressure  of  i  000 
pounds  upon  the  head  of  a  smooth  screw  when  1 1 
turns  cause  the  head  to  advance  two-thirds  of  an 
inch  } 

35.  Find  the  mechanical  advantage  in  a  differential 
screw,  if  the  length  of  the  power  arm  is  2  feet,  and 
there  are  4  threads  to  the  inch  in  the  large  screw, 
and  5  threads  to  the  inch  in  the  small  screw. 

36.  In  a  differential  pulley,  if  the  radii  of  the 
pulleys  in  the  fixed  block  are  as  3  to  2  ;  and  if  the 
weight  of  the  lower  block  is  \\  pounds,  what  weight 
can  be  raised  by  a  force  of  5  pounds } 

37.  In  a  wheel  and  axle  the  diameter  of  the  wheel 
is  7  feet,  of  the  axle  7  inches.     What  weight  can  be 


14  MECHANICS-PROBLEMS. 

raised  by  a  force  of   lo  pounds  acting  at  the  circum- 
ference of  the  wheel  ? 

38.    A  weight  of  448  pounds  is  raised  by  a  cord 

which  passes  round  a  drum  3  feet  in  diameter,  having 

on  its  shaft  a  toothed  wheel  also  3  feet  in  diameter  ; 

a  pinion  8  inches  in  diameter,  and  driven  by  a  winch 

handle    16    inches    long,    gears    with   the 

wheel.     Find  the  power  to  be  applied  to 

the    winch    handle   in    order  to  raise    the 

weight. 

39.  A  tackle  is  formed  of  two  blocks, 
each  weighing  1 5  pounds,  the  lower  one 
being  a  single  movable  pulley,  and  the 
upper  or  fixed  block  having  two  sheaves ; 
the  parts  of  the  cord  are  vertical,  and 
the  standing  end  is  fixed  to  the  movable 
block.  What  pull  on  the  cord  will  sup- 
port 200  pounds  hung  from  the  movable 

block  .!^  and  what  will  then  be  the  pull  on  the  staple 

at  the  upper  block  1 

40.  A  weight  of  400  pounds  is  being  raised  by  a 
pair  of  pulley  blocks,  each  having  two  sheaves  ;  the 
standing  part  of  the  rope  is  fixed  to  the  upper  block, 
and  the  parts  of  the  rope,  whose  weight  may  be  dis- 
regarded, are  considered  to  be  vertical  ;  each  block 
weighs  10  pounds.  What  is  the  pressure  on  the 
point  from  which  the  upper  block  hangs  } 

41.  Two  equal  weights,  each  1 12  pounds,  are  joined 
by  a  rope  which  runs  over  two  pulleys  A  and  B  12 


WORK—  FO  O  T-PO  UNDS.  I  5 

feet  apart  and  in  the  same  horizontal  Hne.  If  a 
weight  of  ten  pounds  is  lowered  on  to  the  rope  half- 
way between  A  and  B  how  far  will  the  rope  deflect  ? 

Work  =  Work 

of  lo-pound  weight  of  two  112-pound  weights. 

42.  A  weight  of  500  pounds,  by  falling  through 
36  feet,  lifts,  by  means  of  a  machine,  a  weight  of  60 
pounds  to  a  height  of  200  feet.  How  many  units  of 
work  has  been  expended  on  friction,  and  what  ratio 
does  it  bear  to  the  whole  amount  of  work  done  t 

43.  The  pull  on  a  tram-car  was  registered  when 
the  car  was  at  the  following  distances  along  the  track  ; 
0,200  pounds;  10  feet,  150  pounds;  25  feet,  160 
pounds  ;  32  feet,  156  pounds;  41  feet,  163  pounds; 
56  feet,  170  pounds;  60  feet,  165  pounds;  73  feet, 
160  pounds.  What  effective  work  was  done  in  pulling 
the  car  through  the  distance  of  73  feet,  and  what 
constant  pull  would  have  produced  the  same  work  ? 

44.  In  lifting  an  anchor  of  i  J  tons  from  a  depth 
of  1 5  fathoms  in  6  minutes,  what  is  the  useful  man- 
power, if  a  man-power  is  defined  as  3  500  foot-pounds 
per  minute } 

45.  Four  hundred  weight  of  material  are  drawn 
from  a  depth  of  80  fathoms  by  a  rope  weighing  i .  1 5 
pounds  per  linear  foot.  How  much  work  is  done 
altogether,  and  how  much  per  cent  is  done  in  lifting 
the  rope.-*  How  many  units  of  33000  foot-pounds 
per  minute  would  be  required  to  raise  the  material  in 
4|  minutes } 


1 6  MECHANICS— PROBLEMS. 


H  O  R  S  E-P  O  W  E  R 

46.  A  gas  engine  must  hoist  3  tons  of  grain  through 
a  vertical  height  of  50  feet  every  minute.  What 
horse-power  must  be  provided  ? 

Work      =  force  X  distance 

of  engine  [minUtC 

=  (3  X  2  000)  pounds  X  50  feet  per 
Now  I  horse-power  =  t^t^  000  foot-pounds  per  minute 

Axr    1            3X2  000  X  £50 
-    .-.Work      =  ^— ^ 

of  engine  ZZ  ^^^ 

—  9tt  horse-power 

47.  A  hod-carrier  who  weighs  1 5  5  pounds  carries 
65  pounds  of  brick  to  the  third  story,  a  vertical  height 
of  20  feet.  How  many  foot-pounds  of  work  has  he 
done  ?  If  he  makes  10  such  trips  in  an  hour,  at  what 
rate  in  horse-power  does  he  work  ? 

48.  A  windmill  raises  by  means  of  a  pump  22  tons 
of  water  per  hour  to  a  height  of  60  feet.  Supposing 
it  to  work  uniformly,  calculate  its  horse-power. 

49.  The  travel  of  the  table  of  a  planing-machine 
which  cuts  both  ways  is  9  feet.  If  the  resistance 
while  cutting  be  taken  at  400  pounds,  and  the 
number  of  revolutions  or  double  strokes  per  hour  be 
80,  find  the  horse-power  absorbed  in  cutting. 

50.  A  forge  hammer  weighing  300  pounds  makes 
100  lifts  a  minute  ;  the  perpendicular  height  of  each 
lift  is  2  feet.  What  is  the  horse-power  of  the  engine 
that  operates  20  such  hammers  ? 


WORK—  HORSE-rO  IVER. 


17 


51.  An  Otto  gas  engine  is  shown  in  the  above 
illustration.  It  has  a  belt  pulley  that  is  36  inches  in 
diameter,  and   makes    150   revolutions   per   minute. 


I  8  MECHANICS— PROBLEMS. 

What  force  for  driving,  shafting,  and  machinery, 
therefore,  can  the  belt  transmit  when  the  engine 
is  developing  its  rated  horse-power  of  twenty- 
one? 

52.  How  many  horse-power  would  it  take  to  raise 
3  hundred  weight  of  coal  a  minute  from  a  pit  whose 
depth  is  66o  feet  ? 

53.  Find  the  horse-power  of  an  engine  which  is  to 
raise  30  cubic  feet  of  water  per  minute  from  a  depth 
of  440  feet. 

54.  Find  the  horse-power  required  to  draw  a  train 
of  100  tons,  at  the  rate  of  30  miles  an  hour,  along  a 
level  railroad,  the  resistance  from  friction  being  16 
pounds  per  ton. 

55.  Each  of  the  two  cylinders  in  a  locomotive 
engine  is  16  inches  in  diameter  and  the  length  of 
crank  is  i  foot.  If  the  driving-wheels  make  105 
revolutions  per  minute,  and  the  mean  effective  steam- 
pressure  is  85  pounds  per  square  inch,  what  is  the 
horse-power } 

56.  The  weight  of  a  train  is  95.5  tons,  and  the 
drawbar  pull  is  6  pounds  per  ton.  Find  the  horse- 
power required  to  keep  the  train  running  at  25  miles 
per  hour. 

57.  A  train,  whose  weight  including  the  engine  is 
100  tons,  is  drawn  by  an  engine  of  150  horse-power ; 
friction  is  14  pounds  per  ton  —  all  other  resistances 
neglected.  Find  the  maximum  speed  which  the 
engine  is  capable  of  maintaining  on  a  level  track. 


WORK  —  HORSE- PO  WER.  \  g 

In  the  electrical  problems  that  follow  observe  that 
I  kilowatt  =  1 ,340  horse-power 
I  horse-power  =  746  watts 

Watts  =  volts  X  amperes 

58.  A  dynamo  is  driven  by  an  engine  that  develops 
230  horse-power.     If  the  efficiency  of  dynamo  is  0.81 » 
what  **  activity  "  in  known  kilowatts  is  represented  by 
the  current  generated  ? 

59.  Electric  lamps  giving  i  candle-power  for  4 
watts  (a)  how  many  10-  and  (d)  how  many  i6-candle 
lamps  may  be  worked  per  electric  horse-power  ?  The 
combined  efficiency  of  engine,  dynamo,  and  gearing 
being  70  per  cent,  what  is  the  candle-power  available 
for  every  indicated  horse-power  ? 

60.  What  electrical  current  expressed  in  amperes 
will  be  used  by  a  250-volt  electric  hoist  when  raising 
2  500  pounds  of  coal  per  minute  from  a  ship's  hold 
1 50  feet  below  dump  cars  on  trestle  work,  the  effi- 
ciency of  the  whole  arrangement  being  50  per  cent  ? 

61.  A  prospective  electric  company  can  find  a 
market  for  900  electrical  horse-power  at  a  city  20 
miles  from  a  suitable  water-power.  Engineers  esti- 
mate losses  in  generating  machinery  10%  ;  in  line 
y% ;  in  transformers  at  load  end  10%  ;  and  the 
efficiency  of  turbines  85%.  The  average  velocity  of 
the  riv^er  is  2  feet  per  second  ;  width  available  near 
dam  40  feet ;  depth  5  feet.  Eind  (a)  the  water- 
power  that  would  be  required  (b)  the  net  fall  that 
proposed  dam  must  afford. 


20 


MECHANICS-PROBL  EMS. 


Fig.   3. 

62.  A  water-motor  is  driven  by  two  jets  i  inch  in 
diameter,  flowing  with  velocity  of  8o  feet  per  second. 
Theoretic  horse-power  would  be  9.9  ;  and  if  efficiency 
of  wheel  is  85  per  cent,  and  the  generator  which 
the  wheel  drives  also  85  per  cent,  what  power  in 
kilowatts  does  the  current  represent  t 

63.  What  is  the  difference  in  tensions  of  the  two 
sides  of  a  30-inch  driving  belt  that  is  running  4  200 
feet  a  minute,  and  transmitting  300  horse-power .? 

In  belt  problems  the  difference  in  tensions  represents  "  force." 

64.  Find  the  speed  of  a  driving-pulley  3.5  feet  in 
diameter  to  transmit  6  horse-power,  the  driving-force 
of  the  belt  being  150  pounds. 


WORK—  IIORSE-PO  WER.  2  I 

65.  A  belt  is  designed  to  stand  a  difference  in 
tension  of  lOO  pounds  only.  Find  the  least  speed  at 
which  it  can  be  driven  to  transmit  20  horse-power. 

66.  A  pulley  3  feet  6  inches  in  diameter,  and  mak- 
ing 150  revolutions  a  minute,  drives  by  means  of  a 
belt,  a  machine  which  absorbs  7  horse-power.  What 
must  be  the  width  of  the  belt  so  that  its  greatest  ten- 
sion may  be  70  pounds  per  inch  of  width,  it  being 
assumed  that  the  tension  in  the  driving-side  is  twice 
that  on  the  slack  side  .-* 

67.  An  endless  cord  stretched  and  running  over 
grooved  pulleys  with  a  linear  velocity  of  3  000  feet 
per  minute,  transmits  5  horse-power.  Find  the  dif- 
ference in  tensions  of  the  cord  in  pounds. 

68.  A  rope  drive  has  a  grooved  pulley  14  feet  in 
diameter  that  makes  30  revolutions  per  minute.  The 
difference  in  tensions  being  100  pounds,  find  the 
horse-power  transmitted. 

69.  A  locomotive  that  can  develop  500  horse- 
power is  drawing  a  train  of  total  weight  100  tons  up 
a  2  per  cent  grade ;  resistances  are  10  pounds  per 
ton.     Find  the  highest  speed  that  can  be  attained. 

Work  =  Work  -f  Work 

of  locomotive  of  resistances  of  lifting  train 

500  X  2>2>  000  =  10  X  iooX^/-f-  100  X  2  000  X  Too  X  '^ 
5  X  33  000  =  10  X  ^-h  20  X  2       X  d 

50  d  =     5   X  33  000 

d  =  3  300  feet  per  minute 
=  37^  miles  per  hour. 


2  2  MECHANICS—  PROBLEMS. 

70.  A  train  of  lOO  tons  weight  runs  at  42  miles 
an  hour  on  a  level  track  ;  resistances  are  8  pounds  per 
ton.  Find  the  speed  of  train  up  a  i  per  cent  grade 
(i  foot  rise  in  100  feet  horizontal)  if  the  engine-power 
is  kept  constant. 

71.  In  1895  a  passenger  engine  on  the  Lake  Shore 
Railroad  made  a  run  of  Z6  miles  at  the  rate  of  73 
miles  an  hour.  Weight  of  train,  250  tons  ;  resistance 
on  level  track,  1 5  pounds  per  ton.  The  engine  was  a 
10- wheeler,  having  drivers  5  feet  8  inches  in  diameter 
and  cylinders  17  X  24  inches.  When  730  horse- 
power was  developed  up  a  i  per  cent  grade  what  was 
the  average  draw-bar  pull  t 

72.  A  98 -horse-power  automobile  has  by  test  in 
Colorado  drawn  a  special  36-ton  locomotive  up  a  12 
per  cent  highway  grade  at  the  rate  of  four  miles  an 
hour.     What  were  the  frictional  resistances  per  ton  1 

73.  A  modern  farming  machine  equipped  with  a 
loo-horse-power  automobile  will  plow,  sow,  and  harrow, 
all  at  the  same  time,  a  strip  30  feet  wide  at  the  rate 
of  3|-  miles  an  hour,  or  80  acres  a  day.  What  force 
is  developed  for  each  foot  width  of  ground  } 

74.  Find  the  total  horse-power  of  two  engines 
which  are  taking  a  train  of  250  tons  down  a  grade  of 
I  in  200  at  60  miles  an  hour,  supposing  the  resistance 
on  the  level  at  this  speed  to  be  35  pounds  a  ton. 


WORK—  HORSE-PO  WEE.  2  3 

75.  An  automobile  that  weighs  5  tons  goes  up  a 
rough  road  of  grade  i  vertical  to  10  horizontal ;  air 
and  frictional  resistances  are  16  pounds  per  ton. 
What  horse-power  must  the  motor  develop  to  main- 
tain a  speed  of  20  miles  an  hour } 

76.  Find  the  horse-power  of  a  locomotive  which  is 
to  move  at  the  rate  of  20  miles  an  hour  up  an  incline 
which  rises  i  foot  in  100,  the  weight  of  the  locomo- 
tive and  load  being  60  tons,  and  the  resistance  from 
friction  12  pounds  per  ton. 

77.  A  steam-crane,  working  at  3  horse-power,  is 
able  to  raise  a  weight  of  10  tons  to  a  height  of  50  feet 
in  20  minutes.  What  part  of  the  work  is  done  against 
friction  t  If  the  crane  is  kept  at  similar  work  for  8 
hours,  how  many  foot-pounds  of  work  are  wasted  on 
friction  .'* 

78.  The  six-master  shown  on  the  next  page  carries 
5  500  tons  of  coal.  It  is  unloaded  by  small  engines 
which  take  up  i  ton  at  each  hoist  ;  average  lift  from 
hold  of  ship  to  top  of  chutes  which  lead  to  cars, 
35  feet;  weight  of  bucket,  i  ton;  2  trips  are  made 
per  minute,  and  25  per  cent  of  power  of  engine  is 
lost  in  friction  and  transmission.  When  two  towers 
are  working  how  long  will  it  take  to  unload  the 
vessel ? 


WORK  —  IIORSE-POWER. 


25 


The  illustration  of  six-master  on  opposite  page  accompanies 
Problem  78. 

79.  An  average  size  coal  barge  will  carry  i  600 
tons.  If  it  is  unloaded  by  two  simple  direct  engines, 
the  coal  being  hoisted  65  feet  to  an  elevated  hopper 
on  the  wharf,  weight  of  bucket  i  ton,  and  carrying  i 
ton  of  coal,  what  horse-power  of  engines  would  be  re- 
quired to  unload  the  i  600  tons  in  20  hours? 

80.  The  locomotive  of  problem  71  made  360.7 
revolutions  per  minute.  What  was  the  mean  effective 
cylinder  pressure } 

Work  =     force  X  distance 
Force  =   ]  tt  1 7-  x  /^  X  2 
Distance  =  2  x  360.7  x  \% 
730  X  ZZ  000  =  (}  It  if  X  /'X  2)  X  (24  X  360.7  X  fl) 


Fig.  4. 


81.   The  engine  shown  in  Fig.  4  has  steam  cylin- 
der 1 5  inches  in  diameter ;  length  of  stroke,  1 5  inches ; 


26 


ME  CHA  NICS-PR  OBLEMS. 


revolutions  per  minute,  275  ;  mean  effective  pressure, 
^6  pounds  per  square  inch.     Find  the  horse-power. 

82.  The  indicator  cards  illustrated  herewith  were 
taken  from  an  engine  of  the  type  shown  in  problem 
81,  diameter  of  steam  cylinder  being  14  inches, 
length  of  stroke  12  inches,  revolutions  per  minute 
300.  Scale  on  cut  the  mean  ordinates,  which  were 
produced  by  indicator  springs  of  stiffness  40  pounds 
to  an  inch,  and  compute  the  indicated  horse-power  of 
the  engine. 


Fig.  5.    Full  Load  Indication. 

83.  The  indicator  cards  shown  below  were  taken 
from  one  of  the  triple-expansion  pumping-engines  at 
the  East  Boston  Station  of  the  Metropolitan  Sewerage. 
The  cards  were  from  two  ends  of  a  high-pressure 
cylinder.  Refer  to  the  cards  and  compute  the  indi- 
cated horse-power.  (A  twenty-four  hours'  duty  trial 
of  this  pumping-engine  was  made  January  17-18, 
1 90 1,  by  engineering  students  of  Tufts  College.) 


IVOR  A'—  IIORSK-rO  WKR. 


27 


Fig.  6.  Head  end.  Card  shown,  one-half  size  ;  area  of  original,  4.69  square 
inches  ;  stiffness  of  spring,  50  pounds  per  square  inch  ;  length  of  stroke, 
30  inches  ;  revolutions  per  minute,  84  ;  piston  diameter,  13^  inches. 


Fig.  7.  Crank  end.  Card  shown,  one-half  size  ;  area  of  original,  4.62  square 
inches  :  stiffness  of  spring,  50  pounds  per  square  inch  ;  length  of  stroke 
30  inches  ;  revolutions  per  minute,  84 ;  piston  diameter,  13^  inches. 

84.  The  average  breadth  of  an  indicator  diagram 
for  one  end  of  a  piston  is  1.58  inches,  and  for  the 
other  end  it  is  1.42  inches,  and  i  inch  represents  32 
pounds  per  square  inch.  Piston,  1 2  inches  diameter  ; 
crank,  i  foot  long;  revokitions  per  minute,  no. 
What  is  the  indicated  horse-power  } 

85.  The  cyHnder  of  a  steam-engine  has  an  internal 
diameter  of  3  feet ;  length  of  stroke,  6  feet  ;  and  it 
makes  10  strokes  per  minute.  Under  what  effective 
pressure  per  square  inch  would  it  have  to  work  in 
order  that  the  piston  may  develop  1 2  5   horse-power  ? 


28  MECHANICS-PROBLEMS, 

The  illustration  of  triple-expansion  engines  on  opposite  p»age 
accompanies  Problem  86. 

86.  Four  pairs  of  triple-expansion  steam-engines 
are  used  to  drive  the  cotton  machinery  of  the  largest 
Fall  River  corporation.  One  of  these  engines  shown 
in  illustration  has  cylinders  26|-  inches  diameter,  36J, 
and  54.  The  steam  pressures  are  :  In  main  pipe,  i  50 
pounds  per  square  inch  ;  in  receiver  between  high  and 
intermediate  cylinders,  40  pounds  ;  in  receiver  between 
intermediate  and  low,  5  pounds.  Vacuum  is  27 
inches.  The  mean  effective  pressures  in  the  cylin- 
ders are  respectively  54  pounds  per  square  inch,  23^- 
and  I2|.  Length  of  stroke  is  5  feet;  piston  speed, 
660  feet  per  minute.      Calculate  the  horse-power. 

87.  An  engine  is  required  to  drive  an  overhead 
traveling  crane  for  lifting  a  load  of  30  tons  at  4  feet 
per  minute.  The  power  is  transmitted  by  means  of 
2j-inch  shafting,  making  160  revolutions  per  minute. 
The  length  of  the  shafting  is  250  feet ;  the  power  is 
transmitted  from  the  shaft  through  two  pairs  of  bevel 
gears  (efficiency  90^;^  each,  including  bearings),  and 
one  worm  and  wheel  (efficiency  85<%,  including  bear- 
ings). Taking  the  mechanical  efficiency  of  the  steam- 
engine  at  80 '}^,  calculate  the  required  horse-power  of 
the  engine. 

88.  An  engine  'working  at  50  horse-power  is 
driven  by  steam  at  75  pounds  pressure  acting  on  pis- 
tons in  two  cylinders.  If  the  area  of  each  piston  is  72 
square  inches,  and  the  length  of  stroke  2  feet,  how 
many  revolutions  does  the  fly-wheel  make  per  minute  ? 


30  MECHANICS-PROBLEMS. 

89.  The  steam-engine  in  use  at  the  Worsted 
Weaving  Mill  of  the  Pacific  Mills  at  Lawrence, 
Mass.,  is  a  Corliss  type  cross-compound  with  steam 
cylinders  19  and  36  inches  diameter;  stroke,  42 
inches;  revolutions,  100  per  minute ;  mean  effective 
pressures,  60  pounds  and  1 3  pounds.  Find  how  many 
looms  weaving  worsted  dress-goods  said  engine  will 
drive,  each  loom  requiring  \  horse-power. 

90.  A  ship  laden  with  coal  must  be  unloaded  at 
the  rate  of  22  tons  of  coal  in  10  minutes.  If  the 
height  of  lift  is  150  feet,  what  horse-power  of  engines 
will  be  required } 

91.  The  fuel  used  in  running  a  steam-engine  is 
coal  of  such  composition  that  the  combustion  of  i 
pound  produces  heat  sufficient  to  raise  the  tempera- 
ture of  12  000  pounds  of  water  1°  Fahr.  It  is  found- 
that  3^^  pounds  of  fuel  are  consumed  per  horse-power 
per  hour.  What  is  the  efficiency  of  the  entire  appa- 
ratus } 

92.  A  steam-engine  uses  coal  of  such  composition 
that  the  combustion  of  i  pound  generates  10  000 
British  thermal  units.  If  40  pounds  of  coal  are  used 
per  hour,  and  if  the  efficiency  is  0.08,  what  horse- 
power is  realized  t 

93.  The  cyhnder  of  a  Corliss-type  steam-engine  is 
30  inches  in  diameter,  stroke  48  inches,  and  it  makes 
85  revolutions  per  minute.  The  steam  pressure  be- 
ing 90  pounds  per  square  inch,  what  is  the  horse- 
power of  the  engine  ? 


WORK  —  HORSE-PO IVER.  3 1 

94.  The  piston  of  a  steam-engine  is  15  inches  in 
diameter  ;  its  stroke  is  2 J  feet,  and  it  makes  20  revo- 
lutions per  minute  ;  the  mean  pressure  of  the  steam 
on  it  is  15  pounds  per  square  inch.  How  many  foot- 
pounds of  work  are  done  by  the  steam  per  minute, 
and  what  is  the  horse-power  of  the  engine  t 

95.  An  engine  has  a  6-foot  stroke,  the  shaft  makes 
30  revolutions  per  minute,  the  average  steam  pres- 
sure is  25  pounds  per  square  inch.  Required  the 
horse-power  when  the  area  of  the  piston  is  i  800 
square  inches,  the  modulus  of  the  engine  being  i  J. 

96.  The  diameter  of  a  steam-engine  cylinder  is  9 
inches  ;  the  length  of  crank,  9  inches  ;  the  number  of 
revolutions  per  minute,  no;  the  mean  effective  pres- 
sure of  the  steam  35  pounds  per  square  inch.  Find 
the  indicated  horse-power. 

97.  The  21  horse-power  gas  engine  of  problem 
51  has  a  12-inch  piston  and  8-inch  crank.  When 
making  150  revolutions  per  minute  with  an  explosion 
every  2  revolutions  what  will  be  the  mean  effective 
pressure  during  a  cycle  } 

98.  The  area  of  a  cross-section  of  the  Charles  River 
at  Riverside,  Massachusetts,  is  408  square  feet.  The 
velocity  of  current  as  found  by  rod  floats  and  current 
meter,  April  17  and  22,  1902,  was  1.12  feet  per  sec- 
ond. What  would  be  the  theoretic  horse-power  of 
this  quantity  of  water  at  the  Waltham  dam,  which 
gives  a  fall  of  12.58  feet  1 


JVOA'A'—  llORSE-PO  WER. 


33 


In  water-power  problems  use 

Work  =  force  X  distance 

(area  X  velocity  X  62J) 

Force  =  pounds  of  water  flowing 
Distance  =  height  of  dam  or  available  fall 

99.  Find  the 
useful  horse-power 
of  a  water-wheel, 
supposing  the 
stream  to  be  loo 
feet  wide  and  5  feet 
deep,  and  to  flow 
with  a  velocity  of 
\  foot  per  second  ; 
the  height  of  the 
fall  is  24  feet,  and 
the  efficiency  of 
tlic  wheel  70  per 
cent. 

100.  A    small 
Fig.  8.--water-Power.                stream    has    mean 

velocity  of  35  feet  per  minute,  fall  of  13  feet  and  a 
mean  section  of  5  feet  by  2.  On  this  stream  is 
erected  a  water-wheel  whose  modulus  is  0.65.  Find 
the  horse-power  of  the  wheel. 

101.  On  page  32  is  shown  the  canal  at  Manchester, 
N.H.,  as  it  passes  the  mills  of  the  Amoskeag  Manufac- 
turing Company.  Width  is  5  i  feet,  depth  of  water  8.9 
feet,  velocity  1.13  feet  per  second.  What  quantity  of 
water  is  flowing  .?  The  height  of  fall  for  the  turbines 
being  27.3  feet,  what  is  the  theoretic  horse-power  ? 


34  MECHANICS-PROBLEMS. 

102.  The  reaction  turbines  of  problem  loi  have 
an  efficiency  of  80  per  cent ;  the  electric  generators, 
90  per  cent.     What  kilowatts  are  available  .-* 

103T  In  winter,  if  2  feet  of  ice  forms  on  this 
canal,  and  the  velocity  drops  to  0.75  feet  per  second, 
and  the  available  fall  becomes  25.0  feet,  what  will 
be  the  kilowatts  available  .!* 

104.  The  mean  section  of  the  Merrimac  Canal  just 
before  it  enters  the  mills  of  the  Merrimac  Manufac- 
turing Company  at  Lowell,  Mass.,  is  48.2  feet  by  10.6 
feet;  mean  velocity  on  Nov.  23,  1901,  was  2.37  feet 
per  second  ;  the  water-wheels  had  a  net  fall  of  35.67 
feet,  and  gave  an  efficiency  of  about  J  J  per  cent.  Find 
the  number  of  broad  looms  weaving  cotton  sheetings 
that  may  be  driven  2^  looms  requiring  one  horse- 
power. 

105.  The  estimated  discharge  of  the  nine  turbines 
at  Niagara  Falls  in  1898  was  430  cubic  feet  per  sec- 
ond for  each  turbine.  The  average  pressure  head  on 
the  wheels  was  that  due  to  a  fall  of  about  136  feet. 
Compute  the  actual  horse-power  available  from  all  tur- 
bines, allowing  an  efficiency  of  82  per  cent. 

106.  The  average  flow  over  Niagara  Falls  is  270  oco 
cubic  feet  per  second.  The  height  of  fall  is  1 6 1  feet. 
In  round  numbers  what  horse-power  is  developed } 

107.  Calculate  the  horse-power  that  can  be  obtained 
for  one  minute  from  an  accumulator  which  makes 


WORK  —  IIORSE-PO  WER. 


35 


one  stroke  in  a  minute  and  has  a  ram  of  20  inches 
diameter,  23  feet  stroke,  loaded  to  a  pressure  of  750 
pounds  per  square  inch. 


Fig.  9.    An  Underwriter  Fire-Pump  with  Standard  Fittings. 

108.  A  fire-pump  for  protection  of  a  50  000-spindle 
cotton-mill  will  deliver  i  000  gallons  of  water  per 
minute  at  100  pounds  pressure.     Large  boiler  capa- 


36  MECHANICS  — PJWBLEMS 

city  is  required  for  such  a  fire-pump  and  for  the  above 
size  150  horse-power  would  be  used.  What  portion 
of  this  boiler  capacity  would  be  required  in  actual 
work  of  delivering  water  ? 

Work  =  force   x  distance 

of  pumping 

Force  =  i  000  x  ^\  pounds  per  minute 

Distance       =  100  x  2.304  feet  head  (i  pound  =  2.304  ft.) 

.-.Work         =  I  000  X  81-  X  100  X  2.304 

of  pumping 

=  I  920  000  foot-pounds  per  minute 
=  58.3  horse-power 

Portion  of  boiler  used  = 

150 
=  0.39,  or  about  one-third 

109.  An  Underwriter  fire-pump  to  protect  an  av- 
erage-sized factory  will  deliver  four  streams  of  water 
through  i|-inch  smooth  nozzles  with  pressure  at  base 
of  play  pipes  of  50  pounds  per  square  inch.  This 
would  correspond  to  a  discharge  of  i  060  gallons  per 
minute.  Loss  of  pressure  through  nozzle  can  be  neg- 
lected ;  and  loss  in  quantity  of  discharge  by  slippage, 
short  strokage,  and  so  on,  will  be  about  10  per  cent. 
Find  the  work  done  by  the  pump. 

110.  A  pump  of  medium  size  used  for  fire  pro- 
tection of  a  factory  will  deliver  three  i|-inch  fire 
streams,  or  750  gallons  per  minute  at  80  pounds 
pressure.  A  boiler  should  be  provided  large  enough 
to  allow  70  per  cent  of  its  capacity  to  remain  as  extra. 
What  should  be  the  nominal  horse-power  of  boiler  } 


WORK—  HORSE-PO  WER.  3  7 

HI.  A  Silsby  steam  fire-engine  delivers  water 
through  a  Siamese  nozzle  that  is  2  inches  in  diameter, 
with  a  pressure  of  80  pounds  per  square  inch  and  a 
mean  velocity  of  106  feet  per  second.  Find  (i)  the 
number  of  cubic  feet  discharged  per  second ;  (2) 
the  weight  of  water  discharged  per  minute  ;  (3)  the 
work  possessed  by  each  pound  of  water  due  to  80 
pounds  pressure ;  (4)  the  horse-power  of  the  engine 
required  to  drive  the  pump,  assuming  the  efficiency 
to  be  70  per  cent. 

112.  At  the  Chestnut  Hill  High-Service  Pumping 
Station  (Boston)  for  the  month  of  October,  1904, 
Kngine  No.  4  pumped  950  780  000  gallons  of  water  ; 
average  lift  was  130.63  feet  ;  total  time  of  pumping, 
744  hours.  What  average  horse-power  was  de- 
veloped } 

113.  The  amount  of  coal  burned  during  the  month 
was  783  148  pounds.  How  many  foot-pounds  of 
work  were  done  for  every  100  pounds  of  coal  burned, 
that  i.s,  what  was  the  Duty  of  the  pumping-engine } 

114.  The  ordinary  fire-engine  when  in  full  opera- 
tion burns  soft  coal,  and  will  consume  in  an  hour  about 
60  pounds  per  fire-stream  of  250  gallons  per  minute. 
Therefore  at  the  70-million  dollar  fire  in  Baltimore, 
February,  1904,  a  500-gallon  engine  that  was  running 
30  hours,  before  the  fire  was  under  control,  consumed 
how  many  pounds  of  coal  } 

115.  Find  the  useful  work  done  each  second  by  a 
fire-engine  which  discharges  water  at  the  rate  of  500 


3  8  MECHANICS  —  PROBLEMS. 

gallons  per  minute  against  a  pressure  of   lOO  pounds 
per  square  inch. 

116.  There  were  6  ooo  cubic  feet  of  water  in  a 
mine  of  6o-fathom  depth  when  a  50-horse-power 
pump  began  to  pump  it  out.  It  took  5  hours  to 
empty  it.  Find  the  number  of  cubic  feet  of  water 
that  ran  into  the  mine  during  the  5  hours,  supposing 
one-fourth  of  the  work  of  the  pump  to  have  been 
wasted. 

117.  Find  the  horse-power  necessary  to  pump  out 
the  Saint  Mary's  Falls  Canal  Lock,  Sault  Ste.  Marie, 
in  24  hours,  the  length  of  the  lock  being  500  feet, 
width  80  feet,  and  depth  of  water  1 8  feet,  the  water 
being  delivered  at  a  height  of  42  feet  above  the  bot- 
tom of  the  lock. 

118.  The  mean  section  of  the  branch  of  the  First 
Level  Canal  at  the  headgates  of  No.  i  Mill,  Whiting 
Paper  Co.,  Holyoke,  Mass.,  is  78  feet  wide  by  14 
deep  ;  from  this  canal  to  the  Second  Level  there  is  a 
fall  of  20  feet,  but  about  2  feet  is  lost  in  penstock  and 
tail-race  ;  velocity  of  flow  in  canal  during  the  daytime 
is  0.20  feet  per  second,  and  the  turbines  that  are 
driven  have  an  efficiency  of  77%.  Find  how  many 
96-inch  Fourdrinier  Paper  Machines  can  be  driven, 
each  machine  requiring  1 00  horse-power. 

119.  What  is  the  horse-power  of  a  stream  that 
passes  through  a  section  of  6  square  feet  at  the  rate 
of  2^  miles  an  hour,  and  has  a  water-fall  of  1 8  feet  ? 


WORK  —  HORSE-PO  WER.  39 

120.  What  horse-power  is  involved  in  lowering  by 
2  feet  the  level  of  the  surface  of  a  lake  2  square  miles 
in  area  in  300  hours,  the  water  being  lifted  to  an 
average  height  of  5  feet  ? 

121.  Taking  the  average  power  of  a  man  as  ^^^th 
of  a  horse-power,  and  the  efficiency  of  the  pump  used 
as  0.4,  in  what  time  will  10  men  empty  a  tank  of 
50  feet  X  30  feet  x  6  feet  filled  with  water,  the  lift 
being  an  average  height  of  30  feet  ? 

122.  A  shaft  560  feet  deep  and  5  feet  in  diameter 
is  full  of  water.  How  many  foot-pounds  of  work  are 
required  to  empty  it,  and  how  long  would  it  take  an 
engine  of  3^^  horse-power  to  do  the  work } 

123.  Required  the  number  of  horse-power  to  raise 
2  200  cubic  feet  of  water  an  hour  from  a  mine  whose 
depth  is  61  fathoms. 

124.  What  weight  of  coal  will  an  engine  of  4  horse- 
power raise  in  one  hour  from  a  pit  whose  depth  is  200 
feet  ? 

125.  A  cut  is  being  made  on  a  4-inch  wrought-iron 
shaft  revolving  at  10  revolutions  per  minute;  the 
traverse  feed  is  0.3  inch  per  revolution;  the  pressure 
on  the  tool  is  found  to  be  435  pounds.  What  is  the 
horse-power  expended  at  the  tool }  How  much  metal 
is  removed  per  hour  per  horse-power  when  the  depth 
of  cut  is  .06  inch,  the  breadth  .06  inch  (triangular 
section)  ? 

ff  OF  THE   ' 

\   UNIVERSITY 


40  MECHANICS-PROBLEMS. 

126.  A  man  rides  a  bicycle  up  a  hill  whose  slope  is 
I  in  20  at  the  rate  of  4  miles  an  hour.  The  weight 
of  man  and  machine  is  iS/^  pounds.  What  work 
per  minute  is  he  doing  ? 

127.  At  the  top  of  the  hill  the  bicyclist  referred  to 
in  example  126  is  met  by  a  strong  head-wind,  and  he 
finds  that  he  has  to  work  twice  as  hard  to  keep  the 
same  rate  of  4  miles  an  hour  on  the  level.  What 
force  is  the  wind  exerting  against  him } 

128.  A  bicyclist  works  at  the  rate  of  one-tenth  of  a 
horse-power,  and  goes  1 2  miles  an  hour  on  the  level. 
Prove  that  the  constant  resistance  of  the  road  is  3.125 
pounds. 

Prove  that  up  an  incline  of  i  vertical  to  50  horizon- 
tal the  speed  will  be  reduced  to  about  5.8  miles  per 
hour,  supposing  that  the  man  and  machine  together 
weigh  168  pounds. 

129.  A  man  rows  a  miles  per  hour  uniformly.  If  R 
pounds  be  the  resistance  of  the  water,  and  P  foot- 
pounds of  useful  work  are  done  at  each  stroke,  find 
the  number  of  strokes  made  per  minute. 

130.  The  resistance  offered  by  still  water  to  the 
passage  of  a  certain  steamer  at  10  knots  an  hour  is 
15  000  pounds.  If  \2''io  of  the  engine  power  is  lost 
by  ''  slip "  —  in  pushing  aside  and  backward  the 
water  acted  on  by  the  screw  or  paddle  —  and  8%  is 
lost  in  friction  of  machinery,  what  must  be  the 
total  horse-power  of  the  engines  t 


WORK— HORSE-POWER.  4 1 

131.  The  United  States  warship  Cokimbia  has  a 
speed  of  23  knots,  with  an  indicated  horse-power  of 
22  000.     Find  the  resistance  offered  to  her  passage. 

132.  The  rise  and  fall  of  the  tide  at  Boston,  Mass., 
is  about  9  feet.  If  the  in-coming  water  for  one 
square  mile  of  ocean  surface  could  be  stored  and  its 
potential  energy  used  during  the  next  6  hours  with 
an  average  fall  of  3  feet,  what  horse-power  would  be 
available  ? 

133.  A  nail  2  inches  long  was  driven  into  a  block 
by  successive  blows  from  a  hammer  weighing  5.01 
pounds  ;  after  one  blow  it  was  found  that  the  head  of 
the  nail  projected  0.8  inches  above  the  surface  of  the 
block ;  the  hammer  was  then  raised  to  a  height  of 
1.5  feet  and  allowed  to  fall  upon  the  head  of  the  nail, 
which,  after  the  blow,  was  found  to  be  0.46  inches 
above  the  surface.  Find  the  force  which  the  hammer 
exerted  upon  the  nail  at  this  blow. 

134.  A  5 00- volt  motor  drives  a  lo-ton  car  up  a 
5  per  cent  grade  at  a  speed  of  12  miles  per  hour  : 
75  per  cent  of  the  work  of  the  motor  is  usefully  ex- 
pended. What  electric  current,  expressed  in  am- 
peres, will  be  required  t 

Work      X  0.75  =  work 

of  motor  of  lifting  car 

135.  The  speed  of  the  "■  Exposition  Flyer  "  on  the 
Lake  Shore  and  Michigan  Southern  Railroad,  when 
running  at  its  maximum,  is  100  miles  per  hour.     At 


42  MECHANICS-PROBLEMS. 

that  speed  what  pull  by  the  engine  would  represent 
one  horse-power?  What  pull  when  running  at  50 
miles  an  hour  ? 

136.  An  express  train  of  weight  250  tons  covers 
40  miles  in  40  minutes.  Taking  the  train  resistances 
on  a  level  track  to  be  20  pounds  per  ton  at  this  speed 
find  the  horse-power  that  engine  must  develop. 

137.  A  train  goes  down  a  grade  of  i  %  for  a  dis- 
tance of  I  mile,  steam  throttle  being  kept  shut ;  it 
then  runs  up  an  equal  grade  with  its  acquired  velocity 
for  a  distance  of  500  yards  before  stopping.  Find 
the  total  resistances,  frictional  or  other,  in  pounds  per 
ton,  which  are  stopping  it. 

138.  A  caboose  and  three  cars  break  away  from  a 
freight  train  and  *' coast  "  down  a  grade  of  2  in  100 
for  a  distance  of  i  mile ;  then  brakes  are  applied  and 
the  cars  stopped  in  200  feet.  Frictional  resistances 
over  whole  distance  being  15  pounds  per  ton,  what 
are  the  brake  resistances  per  ton  t 

Work  =  Work         +  Work 

down  grade  of  friction'  of  brakes 

139.  The  Baltimore  and  Ohio  Railroad  has  now  in  its 
service  (1906)  six  electric  locomotives.  Two  of  recent 
construction  are  used  in  handling  eastbound  freight 
trains  with  steam  locomotives  through  the  city  of 
Baltimore,  which  includes  a  distance  of  about  two 
miles  of  tunnel.  These  locomotives  can  start  and 
accelerate  on  a  level  track  a  train  of  3  000  tons  weight 
with  a  current  consumption  of  2  200  amperes,  which 


WORK— HORSE-  PO  WER.  43 

is  supplied  from  a  power  station  at  560  volts,  but 
reaches  the  locomotives  through  booster  stations  and 
a  storage  battery  at  625  volts.  What  horse-power  do 
they  thus  develop  ? 

140.  These  electric  locomotives  will  draw  on  a  i  % 
grade  a  freight  train  of  i  400  tons  weight  at  10  miles 
an  hour.  Frictional  resistances  being  20  pounds  per 
ton  what  amperes  are  necessary  with  voltage  of  625  ? 


44  MECHANICS-PROBLEMS. 


ENERGY. 

141.  A  train  of  150  tons  is  running  at  50  miles  an 
hour.  What  brake  force  is  required  to  stop  it  in  a 
quarter  of  a  mile  on  a  down  grade  of  2*%,  frictional 
resistance  being  1 5  pounds  per  ton  ? 

Work  +  Work  =  Work      +  Work 

possessed  by  train  gained  in  \  mile  of  brakes  of  resistances 

Work  =  force  X  distance 

possessed  by  train 

Force  =150  tons 

Distance  =  ? 

The  distance  is  found  by  determining  the  vertical  height  that  a 
body  would  have  to  fall  in  order  to  acquire  a  velocity  of  50  miles  an 
hour.  30  miles  an  hour  =  44  feet  per  second.  Therefore  the 
velocity  is  f ^  x  44  =  -f^  feet  per  second.  Now  to  acquire  a 
velocity  of  ^|^  feet  per  second  a  body  would  fall  a  vertical  distance 
that  can  be  found  from  a  fundamental  formula  of  falling  bodies, 

V  =  N/a  gJh  in  which  ^*  varies  for  different  localities 

h  =  84.2  feet. 

That  is,  if  the  train  had  fallen  by  the  action  of  gravity  from  a 

vertical  height  of  84.2  feet  it  would  have  a  velocity  of  ^|^  feet  per 

second,  or  50  miles  an  hour.     Work  can  now  be  analyzed   as  in 

previous  problems. 

Work  =^150  X  84.^  foot-tons 

possessed  by  train 

Work  =  150  X  (t-Io  X^^-V^) 

gained  in  \  mile 

*  The  value  of  g  for  London  is  32.19  feet  per  second  per  second  ;  for  San  Fran- 
isco,  32.15  ;  for  Chicago,  32.16  ;  for  Boston,  32.16.  Practical  limiting  values  for  the 
United  States  are  32.186  at  sea  level  for  latitude  49"^ ;  and  32.089,  latitude  25°  and 
ID  000  feet  above  sea  level.  In  this  book  the  value  32  is  used  for^  so  that  com- 
putations may  be  shortened.  In  many  cases  the  table  on  page  190  will  be  of 
assistance.     The  values  of  \^2gh  there  given  are  based  on  g  as  32.16. 


WORK—  ENERG  Y.  45 

142.  In  the  Westinghouse  brake  tests  (Jan.,  1887), 
at  Weehawken,  a  passenger-train  moving  22  miles  an 
hour  on  a  clown  grade  of  1%  was  stopped  in  91  feet. 
There  was  94%  of  the  train  braked.  Taking  the 
frictional  resistance  as  8  pounds  per  ton,  find  the  net 
brake  resistance  per  ton  on  the  part  of  the  train  that 
was  braked,  and  the  grade  to  which  this  is  equivalent. 

143.  A  freight-car  weighing  20  000  pounds  requires 
a  net  pull  of  10  pounds  per  ton  to  overcome  frictional 
resistance.  If  '*  kicked  "  to  a  level  side  track  with 
velocity  of  10  miles  per  hour,  how  far  will  it  run 
before  stopping } 

144.  A  cake  of  ice  weighing  150  pounds  slides 
down  a  chute  the  height  of  which  is  25  feet;  it 
reaches  the  foot  of  the  shute  with  a  velocity  of  30 
feet  per  second.  During  the  motion  how  many  foot- 
pounds of  energy  must  have  been  lost  t 

145.  A  ship  and  its  cradle  that  weigh  5  000  tons 
slides  down  ways  that  slope  i  foot  in  20  to  the  hori- 
zontal ;  frictional  resistances  amount  to  a  constant 
retarding  force  of  100  tons.  What  v^ould  be  the 
equivalent  height  of  fall  that  would  produce  the  same 
velocity  as  the  ship  possesses  when  she  takes  the 
water  150  feet  down  the  ways } 

146.  If  the  resistance  of  the  water,  anchors,  and 
stop  ropes  amount  to  a  constant  force  of  50  tons, 
how  far  will  the  ship  of  the  preceding  problem  run 
after  she  takes  the  water  ? 


46  MECHANICS-PROBLEMS. 

147.  A  six-inch  rapid-fire  gun  discharges  5  projec- 
tiles per  minute,  each  of  weight  100  pounds,  with 
a  velocity  of  2  800  feet  per  second.  What  is  the 
horse-power  expended  } 

Consider  from  what  vertical  height  a  body  would  fall  to  have  a 
velocity  of  2  Soo  feet  per  second,     {v  ^-=  s/ig/i). 

148.  A  railway  car  of  4  tons,  moving  at  the  rate 
of  5  miles  an  hour,  strikes  a  pair  of  buffers  which 
yield  to  the  extent  of  6  inches.  Find  the  average 
force  exerted  upon  them. 

Work  =  work 

possessed  by  train  by  buffers 

149.  What  is  the  kinetic  energy  of  a  2^ -ton  cable 
car  moving  at  6  miles  per  hour,  loaded  with  36  pas- 
sengers, each  of  average  weight  154  pounds.-*  If 
stopped  in  2  seconds,  what  is  the  average  force  .-^ 

150.  What  is  the  kinetic  energy  of  an  electric  car 
weighing  2|-  tons,  moving  at  10  miles  an  hour,  and 
loaded  with  50  passengers,  of  average  weight  150 
pouilds  } 

151.  The  weight  of  a  ram  is  600  pounds,  and  at 
the  end  of  a  blow  it  has  a  velocity  of  40  feet  per 
second.     What  work  is  done  in  raising  it  } 

152.  Find  the  horse-power  of  a  man  who  strikes 
25  blows  per  minute  on  an  anvil  with  a  hammer  of 
weight  14  pounds,  the  velocity  of  the  hammer  on 
striking  being  32  feet  per  second. 

153.  A  blacksmith's  helper  using  a  16-pound 
sledge  strikes  20  times  a  minute  and  with  a  velocity 
of  30  feet  per  second.      Find  his  rate  of  work. 


WORK— ENERGY.  4^ 

154.  A  ball  weighing  5  ounces,  and  moving  at 
I  000  feet  per  second,  pierces  a  shield,  and  moves  on 
with  a  velocity  of  400  feet  per  second.  What  energy 
is  lost  in  piercing  the  shield  ? 

155.  A  shot  of  I  000  pounds  moving  at  the  rate  of 
I  600  feet  per  second  strikes  a  fixed  target.  How  far 
will  the  shot  penetrate  the  target,  exerting  upon  it  an 
average  pressure  equal  to  a  weight  of  1 2  000  tons  ? 

156.  A  bullet  weighing  i  ounce  leaves  the  mouth 
of  a  r'He  with  a  velocity  of  i  500  feet  per  second.  If 
the  barrel  be  4  feet  long,  calculate  the  mean  pressure 
of  the  powder,  neglecting  all  friction. 

157.  The  bullet  referred  to  in  the  preceding 
problem  penetrates  a  sand  bank  to  the  depth  of  3 
feet.     What  is  the  mean  pressure  exerted  by  the  sand  ? 

158o  An  8-hundred  weight  shot  leaves  a  40-ton  gun 
with  velocity  of  2  000  feet  per  second  :  the  length  of 
the  gun  is  20  feet.  What  is  the  average  force  of 
the  powder  ? 

159.  A  2-ounce  bullet  leaves  the  barrel  of  a  gun 
with  a  velocity  of  i  000  feet  per  second.  Find  the 
work  stored  up  in  the  bullet,  and  the  height  from 
which  it  must  fall  to  acquire  that  velocity. 

160.  What  is  the  kinetic  energy  of  a  5 -hundred 
weight  projectile  fired  with  a  velocity  of  2  000  feet 
per  second .? 

161.  An  8-inch  projectile,  weight  250  pounds, 
strikes  a  sand  butt  going  2  000  feet  per  second,  and 


48  MECHANICS  — PROBLEMS. 

is  stopped  in  25  feet.     If  the  resistance  is  uniform, 
what  is  its  value  in  pounds  ? 

162.  A  hammer  weighing  i  pound  has  a  velocity 
of  20  feet  per  second  at  the  instant  it  strikes  the  head 
of  a  nail.  Find  the  force  which  the  hammer  exerts  on 
the  nail  if  it  is  driven  into  the  wood  ^^  of  an  inch. 

163.  A  fly-wheel  weighs  10  000  pounds,  and  is  of 
such  a  size  that  its  mass  may  be  treated  as  if  concen- 
trated on  the  circumference  of  a  circle  1 2  feet  in  radius. 
What  is  its  kinetic  energy  when  moving  at  the  rate  of 
15  revolutions  a  minute  .-* 

164.  How  many  turns  would  the  above  fly-wheel 
make  before  coming  to  rest,  if  the  steam  were  cut  off, 
and  it  moved  against  a  friction  of  400  pounds  exerted 
on  the  circumference  of  an  axle  i  foot  in  diameter  1 

165.  A  fly-wheel  on  a  21 -horse-power  gas  engine 
of  nominal  speed  1 50  revolutions  per  minute,  must 
store  what  energy  to  provide  for  an  increase  or  de- 
crease in  speed  of  3  revolutions  per  minute  .'* 

166.  The  fly-wheel  of  a  4-horse-power  engine 
running  at  75  revolutions  per  minute  is  equivalent  to 
a  heavy  rim  of  mean  diameter  2  feet  9  inches,  and 
weight  500  pounds.  What  is  the  ratio  of  the  work 
stored  in  the  fly-wheel  to  the  work  developed  in  a 
revolution  .-* 

167.  A  3  horse- power  stamping  machine  presses 
down  once  in  every  2  seconds  ;  its  speed  fluctuates 
from  80  to  1 20  revolutions  per  minute ;  and  to  pro- 
vide for  this  fluctuation  the  fly  wheel  stores  |ths  of 


IVOKK  —  ENERG  V.  49 

all  the  energy  supply  for  2  seconds.     What  energy  is 
thus  stored  per  revolution  ? 

168.  A  nozzle  discharges  a  stream  i  inch  in  diam- 
eter with  a  velocity  of  80  feet  per  second,  (a)  How 
much  work  is  possessed  by  the  water  that  flows  out 
each  minute  .?  (d)  If  this  energy  could  all  be  utilized 
by  a  water-wheel,  what  would  be  its  power } 

169.  Suppose  that  the  above  nozzle  drives  a  water- 
wheel  connected  with  a  pump  which  lifts  water  20 
feet.  If  the  efficiency  of  the  whole  apparatus  is  0.48, 
how  much  water  would  be  lifted  per  minute  .-* 

170.  An  impulse  water-wheel  must  provide  3|  use- 
ful horse-power;  efficiency  of  wheel  is  85%  ;  water- 
pressure  is  60  pounds  per  square  inch  ;  what  size 
nozzle  should  be  used  —  to  the  nearest  eighth  of  an 

inch  ? 

Work        =  force  X  distance 
Force       =  area  X  velocity  X  62^ 

=  area  X  8  V60  x  2.304  x  62^ 
Distance  =  60      x  2.304  feet. 

171.  The  five  streams  shown  on  the  next  page  are 
being  delivered  through  100  feet  of  cotton  rubber- 
lined  hose- with  nozzles  i|  inches  in  diameter.  The 
full  pressure  at  end  of  nozzle  is  50  pounds  per  square 
inch.  What  horse-power  is  the  fire-pump  thus 
delivering .? 

172.  Through  the  100  foot  lines  of  hose  there  is  a 
large  loss  of  pressure.  At  hydrant  the  full  pressure 
is  75  pounds  ;  at  the  nozzle  50  pounds.  What  horse- 
power is  thus  lost } 


Force  illustrated  by  two  fire  streams  being  delivered  by  the  pump  service 
of  the  large  cotton  mills  of  B.B.  &  R.  Knight  at  Natick,  Rhode  Island.  One 
stream  is  being  held  by  men  in  correct  position,  the  other  by  men  who  have 
been  crowded  into  an  awkward  and  dangerous  position.  Pressure  shown  on 
the  gauge  at  the  hydrant  was  75  pounds  per  square  inch. 


FORCES— AT  A    POINT. 


SI 


II.    FORCKS. 

FORCES    ACTING   AT   A    POINT. 

173.  An  acrobat  weighing  1 50  pounds  stands  in 
the  middle  of  a  tight  rope  40  feet  long  and  depresses 
it  5  feet.  Find  the  tension  in  the  rope  caused  by 
his  weight. 


:^^D 


Draw  the  construction  diagram  showing  the  rope,  the  known 
force  of  150  pounds  and  an  arrow  to  indicate  its  direction.  Then 
draw  the  stress  diagram.  Lay  off  150  pounds  parallel  to  the  known 
force  and  at  a  scale  of  i  inch  =  f^o  pounds ;  from  one  end  of  this 
line  ABdrawa  full  line  parallel  with  one  of  the  forces;  from  the 
other  end  a  line  parallel  with  the  other  force.  Complete  the  paral- 
lelogram by  drawing  free  hand  the  dotted  parallel  lines  AD  and  BD. 
Put  arrows  on  forces  beginning  with  the  known  force  of  150  pounds 
—  this  positively  acts  downward  —  then  the  other  arrows  will  follow 
in  order  around  the  full-line  triangle  B  to  C  and  C  to  A.  Thus  A B 
has  become  a  diagonal  of  the  parallelogram  and  is  the  balancing 
force,  or  what  is  more  properly  known  as  the  equilibriant.     If  this 


52 


ME  CHANICS-PR  OBLEMS. 


force  was  acting  in  the  opposite  direction  it  would  be  the  resultant 
of  the  other  two  forces. 

The  full-line  triangle  ABC  constitutes  the  triangle  of  forces.  It 
is  the  keynote  to  the  solution  of  many  problems  in  Forces.  The  mag- 
nitudes and  directions  of  the  forces  can  be  found  by  scaling  from 
this  triangle,  or  by  computations  involving  similar  triangles,  thus, 
geometry  or  trigonometry. 

Forces-At-A-Point  problems  therefore  can  be  solved  as  follows  : 

Draw  a  construction  diagram  showing  dimensions  and  loads. 

Draw  a  stress  diagram. 

First,  the  known  force. 

Complete  the  parallelogram. 

Put  arrows  on  full-line  triangle. 

Scale  or  compute  the  stresses. 

174.  A  Speed-buoy  is  thrown  into  the  water  behind 
a  ship,  and  the  pull  on  the  buoy  by  the  water  is  60 
pounds.  The  two  ropes  that  connect  the  buoy  with 
the  ship  make  an  angle  of  15°  at  their  point  of  attach- 
ment.    Find  the  stresses  on  the  ropes. 

175.  Two  men  pull  a  body  horizontally  by  means 
of  ropes.  One  exerts  a  force  of  28  pounds  directly 
north,  the  other  a  force  of  42  pounds  in  direction  N. 
42°E.  What  single  force  would  be  equivalent  to 
the  two } 

176.  Three  cords  are  knotted  together  ;  one  of 
these  is  pulled  to  the  north  with  a  force  of  6  pounds, 
another  to  the  east  with  a  force  of  8  pounds.  With 
what  force  must  the  third  be  pulled  to  keep  the  whole 
at  rest } 

177.  Two  persons  lifting  a  body  exert  forces  of  44 
pounds  and  60  pounds  on  opposite  sides  of  the  ver- 


FORCES— AT  A    POINT.  53 

tical,  but  each  with  an  indination  of  28°.    What  single 
force  would  produce  the  same  effect  ? 

178.  A  force  of  50  units  acts  along  a  line  inclined 

at  an  angle  of  30°  to  the  horizon.     Find,  by  construe-        c 
tion   or  otherwise,  its    horizontal    and    vertical    com- 
ponents. 

179.  Explain  the  boatman's  meaning  when  he  says 
that  greater  force  is  developed  when  a  mule  hauls  a 
canal  boat  with  a  long  rope  than  with  a  short  one. 
Is  the  same  true  of  a  steam-tug  when  towing  a  four- 
master  } 

180.  Two  strings,  one  of  which  is  horizontal,  and 
the  other  inclined  to  the  vertical  at  an  angle  of  30°, 
support  a  weight  of  10  pounds.  Find  the  tension  in 
each  string. 

181.  Two  forces  of  20  pounds,  and  one  of  2 1  act 

at  a  point.     The  angle  between  the  first  and  second        * 
is    120°,  and   between    the    second    and    third,    30°. 
Find  the  resultant. 

182.  Forces  of  9  pounds,  12,  13,  and  26,  act  at  a 
point  so  that  the  angles  between  the  successive 
forces  are  equal.     Find  their  resultant. 

183.  A  weightless  rod,  3  feet  long,  is  supported 
horizontally,  one  end  being  hinged  to  a  vertical  wall, 
and  the  other  attached  by  a  string  to  a  point  4  feet  ^ 
above  the  hinge;  a  weight  of  120  pounds  is  hung 
from  the  end  supported  by  the  string.  Calculate  the 
tension  in  the  string,  and  the  pressure  along  the  rod. 


5  4  MECHANICS-PROBLEMS. 

184.  A  weight  of  loo  pounds  is  fixed  to  the  top  of 
a  weightless  rod  or  strut  5  feet  long  whose  lower  end 
rests  in  a  corner  between  a  floor  and  a  vertical 
wall,  while  its  upper  end  is  attached  to  the  wall  by  a 
horizontal  wire  4  feet  long.  Calculate  the  tension  in 
the  wire,  and  the  thrust  in  the  rod. 

185.  A  rod  AB  is  hinged  at  A  and  supported  in 
a  horizontal  position  by  a  string  BC  making  an 
angle  of  45°  with  the  rod  ;  the  rod  has  a  weight  of 
10  pounds  suspended  from  B.  Find  the  tension  in 
the  string  and  the  force  at  the  hinge.  (The  weight 
of  the  rod  can  be  neglected.) 

186.  A  simple  triangular  truss  of  30  feet  span  and 
5  feet  depth  supports  a  load  of  4  tons 
at  the  apex.  Find  the  forces  acting  on 
rafters  and  tie  rod. 

187.    A  derrick    is    set    as  shown    in 
sketch,  the  load  being  8  tons.     Find  the 
Fig.  12.         stress  in  the  boom  and  the  tackle. 

188.  A  stiff -leg  steel  derrick,  with  mast  55  feet 
high,  boom  85  feet  long,  set  with  tackle  40  feet  long, 
as  shown  in  cut,  is  raising  two  boilers  of 
50  tons  weight.  Find  stresses  in  boom 
and  tackle.     (See  illustration  on  page  55.) 

189.  Find    the    stress    in    tackle    and 
compression    in   boom  of  towers  for  six- 
master  shown  on  page  24  when  bucket, 
weighing  with  its  load  2  tons,  is  set  in  position  shown 
by  Fig.  13. 


FORCES— AT  A    POJXT. 


55 


Fig.  14. 

190.  A  balloon  capable  of  raisin<T^  a  weight  of  360 
pounds  is  held  to  the  ground  by  a  rope  which  makes 
an  angle  of  60°  with  the  horizon.  Determine  the 
tension  of  the  rope  and  the  horizontal  pressure  of  the 
wind  on  the  balloon. 

191.  A  uniform  beam    10  feet  long,  weighing  80 
pounds,  is  suspended  from  a  horizontal  ceiling  by  two 
strings   attached  at   its    ends,  and  at  points   16  feet       ^ 
apart  in  the  ceiling.      Find  the  tension  in  each  string. 


192.  A  boat  is  towed  along  a  canal  50  feet  wide, 
by  mules  on  both  banks  ;  the  length  of  each  rope 
from  its  point  of  attachment  to  the  bank  is  72  feet : 


56  MECHANICS-PROBLEMS. 

the  boat  moves  straight  down  the  middle  of  the 
canal.  Find  the  total  effective  pull  in  that  direction, 
when  the  pull  on  each  rope  is  800  pounds. 

193.  A  boat  is  being  towed  by  a  rope  making  an 
angle  of  30°  with  the  boat's  length  ;  the  resultant 
pressure  of  the  water  and  rudder  is  inclined  at  60° 
to  the  length  of  the  boat,  and  the  tension  in  the  rope 
is  equal  to  the  weight  of  half  a  ton.  Find  the  re- 
sultant force  in  the  direction  of  the  boat's  length. 

194.  In  a  direct-acting  steam-engine  the  piston- 
pressure  is  22  500  pounds  ;  the  connecting-rod  makes 
a  maximum  angle  of  15°  with  the  line  of  action  of  the 
piston.     P'ind  the  pressure  on  the  guides. 

195.  A  man  weighing  160  pounds  sits  in  a  loop  at 
the  end  of  a  rope  10  feet  3  inches  long,  the  other 
end  being  fastened  to  a  point  above.  What  horizon- 
tal force  will  pull  him  2  feet  3  inches  from  the  verti- 
cal, and  what  will  then  be  the  pull  on  the  rope } 

196.  A  man  weighing  160  pounds  sits  in  a  ham- 
mock suspended  by  ropes  which  are  inclined  at  30° 
and  45°  to  vertical  posts.     Find  the  pull  in  each  rope. 

197.  Two  equal  weights,  W,  are 
attached  to  the  extremities  of  a 
flexible  string  which  passes  over 
w  three  tacks  arranged  in  the  form 
of  an  isosceles  triangle  with  the 
base  horizontal,  the  vertical  angle  at  the  upper  tack 
being  1 20°.     Find  the  pressure  on  each  tack. 


y 


FORCES— AT  A    POINT.  5/ 

198.  A  rod  AB  5  feet  long,  without  weight,  is 
hung  from  a  point  C  by  two  strings,  which  are  at- 
tached to  its  ends  and  to  the  point ;  the  string  AC  is 
3  feet  long,  and  the  string  BC  2  feet ;  a  weight  of 
2  pounds  is  hung  from  A  and  a  weight  of  3  pounds 
from  B.  Find  the  tension  of  the  strings  and  the 
condition  that  these  may  be  in  equilibrium. 

199.  A  weight  of  10  pounds  is  suspended  by  two 
strings,  7  and  24  inches  long,  the  other  ends  of 
which  are  fastened  to  the  extremities  of  a  rod  25 
inches  in  length.  Find  the  tension  of  the  strings 
when  the  weight  hangs  immediately  below  the  middle 
point  of  the  rod. 

200.  AB  is  a  wall,  and  C  a  fixed  point  at  a  given 
perpendicular  distance  from  it  ;  a  uniform  rod  of 
given  length  is  placed  on  C  with  one  end  against  AB. 
If  all  the  surfaces  are  smooth,  find  the  position  in 
which  the  rod  is  in  equilibrium. 

201.  AB  is  a  uniform  beam  weighing  300  pounds. 
The  end  A  rests  against   a  smooth  verti-  ^ 
cal  wall,  the  end  B  is  attached  to  a  rope 
CB.       Point    C    is    vertically    above    A,  X 
length  of  beam  is  4    feet,    rope    7  feet. 
Represent  the  forces  acting,  and  find  the  X|^ 
pressure  against  the  wall  and  the  tension  ^' 
in  the  rope.  Fig.  jg. 

202.  A  wagon  weighing  2  200  pounds  rests  on  a 
slope  of  inclination  30°.  What  are  the  equivalent 
forces  parallel  and  perpendicular  to  the  plane  ? 


/ 


58  MECHANICS-PROBLEMS. 

203.  AB  is  a  rod  that  can  turn  freely  round  one 
end  A ;  the  other  end  B  rests  against  a  smooth  in- 
clined plane.  In  what  direction  does  the  plane  react 
upon  the  rod  }  Illustrate  your  answer  by  a  diagram 
showing  the  rod,  the  plane,  and  the  reaction. 

204.  A  wagon  weighing  2  tons  is  to  be  drawn  up 
a  smooth  road  which  rises  4  feet  vertically  in  a  dis- 
tance of  32  feet  horizontally  by  a  rope  parallel  to  the 
road.  What  must  the  pull  of  the  rope  exceed  in 
order  that  it  may  move  the  wagon  t 

205.  What  weight  can  be  drawn  up  a  smooth  plane 
rising  i  in  5  by  a  pull  of  200  pounds  {a)  when  the 
pull  is  parallel  with  the  plane  t  (b)  when  it  is  hori- 
zontal .? 

206.  A  horse  is  attached  to  a  dump-car  by  a  chain, 
which  is  inclined  at  an  angle  of  45°  to  the  rails; 
the  force  exerted  by  the  horse  is  672  pounds.  What 
is  the  effective  force  along  the  rails  .? 

~  207.  The  angle  of  inclination  of  a  smooth  inclined 
plane  is  45°  :  a  force  of  3  pounds  acts  horizontally, 
and  a  force  of  4  pounds  acts  parallel  to  the  plane. 
Find  the  weight  which  they  will  be  just  able  to 
support. 

208.  A  body  rests  on  a  plane  of  height  3  feet,  length 
5  feet.  If  the  body  weighs  14  pounds,  what  force  act- 
ting  along  the  plane  could  support  it,  and  what  would 
be  the  pressure  on  the  plane  } 


FORCES— AT  A   POINT.  50 

209.  A  number  of  loaded  trucks  each  containing 
one  ton,  standing  on  a  given  part  of  a  smooth  tram- 
way, where  the  inclination  is  30°,  support  an  equal 
number  of  empty  trucks  on  another  part,  where  the 
inclination  is  45°.     Find  the  weight  of  a  truck. 

210.  Two  planks  of  lengths  7  yards  and  6  yards 
rest  with  one  end  of  each  on  a  horizontal  plane,  the 
other  ends  in  contact  above  that  plane  ;  two  weights 
are  supported  one  on  each  plank,  and  arc  connected 
by  a  string  passing  over  a  pulley  at  the  junction  of 
the  planks  ;  the  weight  on  the  first  plank  is  2 1  pounds. 
What  is  the  weight  on  the  other,  friction  not  being 
considered  'i 

211.  The  weight  of  a  wheel  with  its  load  is  2  tons, 
diameter  of  wheel  5  feet.  Find  the  least  horizontal 
force  necessary  to  pull  it  over  a  stone  4  inches  high. 
(When  the  wheel  begins  to  rise  three 
forces  are  acting  :  P,  W,  and  R  the 
reaction.     It  is  required  to  find  P.) 

212.  A  rectangular  box,  contain- 
ing  a  200-pound  ball,   stands  on  a  ^^^'  '7- 
horizontal  table,  and  is  tilted  about  one  of  its  lower 
edges  through  an  angle  of  30.°     Find  the  pressure  be- 
tween the  ball  and  the  box. 

213.  An  iron  sphere  weighing  50  pounds  is  resting 
against  a  smooth  vertical  wall  and  a  smooth  plane 
which  is  inclined  60°  to  the  horizon.  Find  the  pres- 
sure on  the  wall  and  plane. 


6o  MECHANICS-PROBLEMS. 

214.  A  beam  weighing  400  pounds  rests  with  its 
ends  on  two  inclined  planes  whose  angles  of  inclina- 
tion to  the  horizontal  are  20°  and  30°.  Find  the 
pressures  on  the  planes. 

215.  A  thread  14  feet  long  is  fastened  to  two 
points  A  and  B  which  are  in  the  same  horizontal  line 
and  10  feet  apart ;  a  weight  of  25  pounds  is  tied  to 
the  thread  at  a  point  P  so  chosen  that  AP  is  6  feet  — 
therefore  BP  is  8  feet  long.  The  weight  being  thus 
suspended,  find  by  means  of  construction  or  otherwise, 
what  are  the  tensions  of  the  parts  AP  and  BP  of  the 
thread. 

216.  AC  and  BC  are  two  threads  4  feet  and  5  feet 
long,  respectively,  fastened  to  fixed  points  A  and  B,  . 
which  are  in  the  same  horizontal  line  6  feet  apart ;  a 
weight  of  50  pounds  is  fastened  to  C.  Find,  by 
means  of  a  line  construction  drawn  to  scale,  the  pull 
it  causes  at  the  points  A  and  B.  Each  of  the  threads 
AC  and  BC  is,  of  course,  in  a  state  of  tension. 
What  are  the  forces  producing  the  tension  1 

217.  A  boiler  weighing  3  000  pounds  is   supported 

by  taqkles  from  the    fore   and  main  yards.     If  the      ^ 
tackles  make  angles  of  25°  and  35°  respectively  with 
the  vertical,  what  is  the  tension  of  each  .'* 

218.  A  piece  of  wire  26  inches  long,  and  strong 
enough  to  support  directly  a  load  of  100  pounds, 
is  attached  to  two  points  24  inches  apart  in  the  same 
horizontal  line.     Find  the  maximum  load  that  can  be 


FORCES— AT  A    POINT.  6 1 

suspended  at  the  middle  of  the  piece  of  wire  without 
breaking  it. 

219.  A  picture  of  50  pounds  weight  hanging  ver- 
tically against  a  smooth  wall  is  supported  by  a  string 
passing  over  a  smooth  hook  ;  the  ends  of  the  string 
are  fastened  to  two  points  in  the  upper  rim  of  the 
frame,  which  are  equidistant  from  the  center  of  the 
rim,  and  the  angle  at  the  peg  is  60°.  Find  the  tension 
in  the  string. 

220.  A  weight  W  attached  by  two  connecting 
cords  of  lengths  a  and  b  to  two  fixed  points  A  and  B, 
and  separated  by  a  horizontal  interval  r,  are  in  equilib- 
rium under  the  action  of  gravity.  Required  the 
stresses  P  and  O  in  the  cords. 

221.  Two  equal  rods  AB  and  BC  are  loosely  jointed 
together  at  B.  C  and  A  rest  on  two  fixed  supports 
in  the  same  horizontal  line,  and  are  connected  by  a 
cord  equal  in  length  to  AB.  If  a  weight  of  12  pounds 
be  suspended  from  B,  what  is  the  pressure  produced 
along  AB  and  BC,  and  the  tension  in  the  cord  } 

222.  Two  spars  are  lashed  together  so  as  to  form 
a  pair  of  shears  as  shown  in  sketch. 
They  stand  with  their  '*  heels  "  20  feet 
apart,  and  would  be  40  feet  high  when 
vertical.  What  is  the  tension  in  the 
guy  and  thrust  in  the  legs  when  a  load 
of  30  tons  is  being  lifted  } 

Suppose  that  a  single  leg    should  replace  the  Fig.  18. 


62 


MECHA  NICS-PROBLEMS, 


two  spars.  The  stress  can  easily  be  found  in  this  imaginary  leg  by 
considering  that  at  A,  in  this  plane,  three  forces  meet,  —  the  imagi- 
nary leg,  the  back  guy,  and  the  vertical  load  of  30  tons.  Then 
consider  the  three  forces  at  A  in  the  plane  of  the  legs,  and  thus 
find  the  stresses  in  the  two  equal  spars. 


223.    When  the  spars  become  vertical  what  stresses 
will  exist  for  the  load  of  30  tons  ? 


Fig.  19. 


224.  Figs.  19-20  show  a 
pair  of  shears  erected  at 
Sparrow's  Point,  Md.,  for 
the  Maryland  Steel  Com- 
pany. The  two  front  legs 
are  hollow  steel  tubes  116 
feet  long,  and  inclined  35 
feet  out  of  the  vertical. 
The  back  leg  is  126  feet  long,  and  is  connected  to 
hydraulic  machines  for  operating  the  shears.  How 
much  are  the  forces  acting  in  these  legs  when  a 
Krupp  gun  weighing  122  tons  is  being  lifted.'* 

225.  Each  leg  of  a  pair  of  shears  is  50  feet  long. 
They  are  spread  20  feet  at  the  foot.  The  back  stay 
is  75  feet  long.  Find  the  forces  acting  on  each 
member  when  lifting  a  load  of  20  tons  at  a  distance  of 
20  feet  from  the  foot  of  the  shear  legs,  neglecting  the 
weight  of  structure. 

226.  Shear  legs  each  50  feet  long,  30  feet  apart  on 
horizontal  ground,  meet  at  point  C,  which  is  45  feet 
vertically  above  the  ground  ;  stay  from  C  is  inclined 


FORCES— AT  A   POINT. 


63 


Fig.  20. 

at  40°  to  the  horizon  ;  a  load  of  10  tons  hangs  from 
C.     Find  the  force  in  each  leg  and  stay. 

227.  A  vertical  crane  post  is  10  feet  high,  jib  30 
feet  long,  stay  24  feet  long,  meeting  at  a  point  C. 
There  are  two  back  stays  making  angles  of  45°  with 
the  horizontal ;  they  are  in  planes  due  north  and  due 


FORCES  — AT  A    POINT. 


65 


west  from  the  post.  A  weight  of  5  tons  hangs  from 
C.  Fmd  the  forces  in  the  jib  and  stays —  ist,  when 
C  is  southeast  of  the  post ;  2d,  when  C  is  due  east ; 
3d,  when  C  is  due  south. 

228.  The  view  on  opposite  page  shows  one  of  the 
largest  dipper  dredges  ever  built,  the  "  Pan  American," 
constructed  at  Buffalo  in  1899  for  use  on  the  Great 
Lakes.  An  A-frame,  the  legs  of  which  are  57  feet 
long  and  40  feet  apart  at  the  bottom,  is  held  at  the 
apex  by  four  cables  which  are  100  feet  long.  The 
boom  is  53  feet  long  and  weighs  30  tons.  The 
handle,  which  weighs  about  4  tons,  is  60  feet  long, 
and  carries  on  its  end  a  dipper  weighing  16  tons, 
which  will  dredge  up  8 J  cubic  yards,  or  about  12 
tons,  of  material  at  one  load. 

The  dipper  is  operated  by  a  wire  rope  passing  over 
a  pulley  on  the  outward  end  of  the  boom.  In  the 
position  represented  by  the  outline  sketch,  the  boom 
is  inclined  to  the 


water  surface  at 
an  angle  of  30°, 
the  dipper  is  car- 
rying the  full 
load,  and  the  han- 
dle is  in  a  hori- 
zontal position 
with  its  middle  point  supported  at  a  point  on  the 
boom  23  feet  from  the  foot  of  the  boom.  The  apex 
of  the  A-frame  is  vertically  above  the  foot  of  the 
boom.     Compute    the  forces  acting  in  the   100-foot 


Fig.  21. 


V 


66  MECHANICS-PROBLEMS. 

back-Stays  (considering  them  to  be  one  rope,  in  posi- 
tion as  per  sketch),  in  the  legs  of  the  A-frame,-  in  the 
boom,  and  in  the  wire  rope  which  raises  the  dipper. 

229-  A  tripod  whose  vertex  is  A,  and  whose  legs 
are  AB,  AC,  AD,  of  lengths  8  feet,  8.5,  and  9  re- 
spectively, sustains  a  load,  of  2  tons.  The  ends 
B,  C,  D,  form  a  triangle  whose  sides  are  BC  7  feet, 
CD  6  feet,  BD  8  feet.     Find  the  stress  in  each  leg. 

Sketch  the  figure  and  put  on  the  dimensions.  Then  draw  to  scale 
the  base  BCD,  and  in  this  horizontal  plane  locate  the  vertices  A", 
A-',  and  h!"  of  the  three  faces  of  the  pyramidal-shaped  figure  that 
is  formed  by  the  legs  of  the  tripod.  Perpendiculars  drawn  from  A', 
A"  and  A"'  to  their  respective  sides  of  the  triangle  BCD  will  locate 
at  their  intersection  the  projection  of  vertex  A.  Now  pass  a  vertical 
plane,  for  example,  through  AB  and  the  load  of  2  tons ;  note  the  in- 
tersection E  with  line  CD.  AE  can  be  considered  as  an  imaginary 
leg,  and  the  stress  in  it  can  be  graphically  determined  as  hereto- 
fore, also  the  stress  in  AC  and  AD. 

30.  A  tripod  with  8-foot  legs  is  to  be  used  for 
lowering  a  2-ton  water-pipe.  How  far  apart  can  the 
bottoms  of  legs  be  spread,  if  in  an  equilateral  triangle, 
so  that  not  over  i  ton  stress  will  come  on  each  leg } 

231.  A  chandelier  of  weight  500  pounds  is  to  hang 
under  the  middle  of  a  triangle  12  feet  X  8  X  8.  Two 
of  the  chains  are  to  be  20  feet  long.  What  should 
be  the  length  of  the  third  chain }  What  stresses 
would  exist  in  chains  t 

232.  ABCD  is  a  square  ;  forces  of  i  pound,  6,  and 
9  act  in  directions  AB,  AC,  and  AD  respectively. 
Find  the  magnitude  of  their  resultant. 


FORCES— -AT  A    POINT.  6/ 

233.  A,  B,  C,  D,  are  the  angular  points  of  a  square 
taken  in  order  ;  three  forces  act  on  a  particle  at  A, 
viz.  one  of  7  units  from  A  to  B,  a  second  of  10  units 
from  D  to  A,  and  a  third  of  5  V2  units  along  the 
diagonal  from  A  to  C.  Find,  by  construction  or 
otherwise,  the  resultant  of  these  three  forces. 

234.  Forces  P,  2P,  3P,  and  4P  act  along  the  sides 
of  a  square  A,  B,  C,  D,  taken  in  order.  Find  the 
magnitude,  direction,  and  line  of  action  of  the  result- 
ant. 


235.  A  sinker  is  attached  to  a  fishing-line  which  is 
then  thrown  into  running  water.  Show  by  means  of 
a  diagram  the  forces  which  act  on  the  sinker  so  as  to 
maintain  equilibrium. 

236.  A  uniform  rod  6  feet  long,  weighing  10  pounds, 
is  supported  by  a  smooth  pin  and  by  a  string  6  feet 
long  which  is  attached  to  the  rod  i  foot  from  one 
end  and  to  a  nail  vertically  above  the  pin,  4  feet  dis- 
tant. Show  by  construction  the  position  in  which 
the  rod  will  come  to  rest. 


237.  A  light  rod  AB  can  turn  freely  round  a  hinge 
at  A  ;  it  rests  in  an  inclined  position  against  a  smooth 
peg  near  the  end  B  ;  a  weight  is  hung  from  the  middle 
of  the  rod.  Show  in  a  diagram  the  forces  which 
keep  the  rod  at  rest,  and  name  them. 


^/ 


y 


68 


ME  CIIA  NICS-PR  OBLEMS. 


Fig.    22. 


238.  A  weight  W  on  a  plane 
inclined  30°  to  the  horizontal  is 
supported  as  shown  in  cut.  The 
angles  ^  being  equal.  Find  the 
ratio  of  the  power  to  the  weight. 

239.  Discuss  the  action  of  the 
wind  in  propelling  a  sailing-vessel. 

Let  AB  be  the  keel,  CD  the  sail.  Let  the 
force  of  the  wind  be  represented  in  magnitude  /s,. 
and  direction  by  EF.  The  component  GF 
of  EF,  perpendicular  to  the  sail,  is  the  effec- 
tive component  in  propelling  the  ship ;  the 
other  component  EG,  parallel  to  the  sail,  is 
useless ;  but  GF  drives  the  ship  forward  and 
sidewise.  The  component  GH  of  GF,  perpendicular  to  AB,  pro- 
duces side  motion,  or  leeway  ;  and  the  other  component  HF,  along 
the  keel,  produces  forward  motion,  or  headway. 

240.  A  sailing-boat  is  being 
driven  forward  by  a  force  of  300 
pounds  as  shown  in  Fig.  24. 
What  force  is  P  acting  in  direction 
of  motion  of  the  boat } 

241.  Discuss  the  action  of  the 
rudder  of  a  vessel  in  counteracting 

leeway.     Show  that  one  effect  of  the  action  of  the 
rudder  is  to  diminish  the  vessel's  motion. 


Fig.  24. 


242.    A  thread  of  length  /  has  its  ends  fastened  to 
two  points  in  a  line  of  length  c,  and  inclined  to  the 
vertical  with  angle  0  ;  a  weight  W  hangs  on  the  thread       ^ 
by  means  of  a  smooth  hook.     Find  the  position  in 


FORCES.  — AT  A  POINT.  69 

which  the  weight  comes  to  rest  and  the  tension  in  the 
thread. 

243.  A  smooth  ring  weighing  40  pounds  slides 
along  a  cord  that  is  attached  to  two  fixed  points  in  a 
horizontal  line.  The  distance  between  the  points 
being  one-half  length  of  cord,  find  position  in  which 
weight  will  come  to  rest  and  the  tension  in  the  string 
near  the  points  of  attachment. 

244.  A  small  heavy  ring  A,  which  can  slide  upon 
a  smooth  vertical  hoop,  is  kept  in  a  given  position  by 
a  string  AB,  B  being  the  highest  point  of  the  hoop. 
Show  that  the  pressure  between  the  ring  and  the 
hoop  is  equal  to  the  weight  of  the  ring. 

245.  Draw  a  figure  showing  the  mechanical  con- 
ditions of  equilibrium  when  a  uniform  beam  rests  with 
one  extremity  against  a  smooth  vertical  wall,  and  the 
other  inside  a  smooth  hemispherical  bowl. 

246.  A  ball  8  inches  in  diameter,  weighing  100 
pounds,  rests  on  a  plane  inclined  30°  to  the  horizon, 
and  is  held  in  equilibrium  by  a  string  4  inches  long 
attached  to  a  sphere  and  to  an  inclined  plane.  Rep- 
resent the  forces  acting,  and  find  their  values. 

247.  A  uniform  sphere  rests  on  a  smooth  inclined 
plane,  and  is  held  by  a  horizontal  string.  To  what 
point  on  the  surface  of  the  sphere  must  the  string  be 
attached  }    Draw  a  figure  showing  the  forces  in  action, 


70 


ME  CHA  NIC S- PR  OBL  EMS. 


248.  A  uniform  bar  of  weight  20  pounds,  length  12 
feet,  rests  with  one  end  inside  a  smooth  hemispheri- 
cal bowl,  and  is  supported  by  the  edge  of  the  bowl 
with  2  feet  of  the  bar  outside  of  it.  Draw  the  forces 
producing  equilibrium,  and  find  their  values. 


The  stresses  in  a  roof  or  bridge  truss  that  carries  a  uniform  load 
are  best  determined  by  finding  in  place  of  the 
uniform  loads  equivalent  apex  loads.  And  a 
fact  that  is  often  obscure  to  students  is,  that  a 
part  of  this  uniform  load  is  not  included  in 
our  computation  of  stresses.  In  the  truss  of 
Fig.  25  the  portions  a  of  uniform  load  are  not 
included  in  the  compressive  stresses  of  A  C  and 
C  B.  This  fact  will  be  further  understood  by 
solving  the  problems  that  follow. 


Fig.  25. 


249.  Two  floor  beams  of  16  feet  length  meet  at 
a  post,  Fig.  26.  The  load,  10  feet  width  of  bay  for 
each  beam,  is  150  pounds  per  square  foot.  What  will 
be  the  load  carried  by  the  post  }  If  it  is  found  that 
the  post  must  be  removed  so  as  to  give  better  floor 
space  the  plan  of  Fig.  27  could  be  used.  What 
would  then  be  the  stress  in  the  short  post  (3  feet 
long),  and  in  the  two  rods,  and  in  the  floor  beams  } 


Fig.  26.  Fig.  37. 

250.    Now  if   the  same  conditions  exist  as  in  the 
preceding  problem,  except  that  the  rods,  instead  of 


FORCES—  AT  A    POINT. 


71 


being  fastened  to  the  ends  of  the  beam  are  fastened 
to  straps  on  the  outside  of  the  wall,  what  will  then 
be  the  stresses  in  post,  rods,  and  floor  beam  ? 

251.  The  slopes  of  a  simple  triangular  roof-truss 
are  30^  and  45°,  and  the  span  is  50  feet.  The  trusses 
are  set  10  feet  apart,  and  the  weight  of  the  roof  cov- 
ering and  snow  is  50  pounds  per  square  foot  of  roof. 
I-^ind  the  stresses  in  tie-rod  and  rafters. 


V/Vi 
10  >  <  ip;>  yv) 


m 


11 


■ft^ 


f 


c  — < 


Plan 
Fig.  a8. 


Elevation 
Fig.  29. 


The  load  on  any  truss  would  be  represented  by  the  shaded  area 
in  Fig.  28.  Find  this  load  and  then  the  apex  loads  A,  C,  and  B,  and 
observe  that,  according  to  explanation  of  preceding  problems,  the 
loads  A  and  B  do  not  enter  into  our  computations.  C  alone  is  re- 
quired. Having  found  C,  the  stresses  in  rafters  can  be  determined. 
Then  find  the  stress  that  each  rafter  transmits  to  the  tie-rod. 


252.    In  a  roof  of    32  feet  span 
and  height  12  feet  the  trusses  are         1  b 
10   feet    apart,    and    the   members  *  a  ^  f 
EF,  GH,  come  to  the  middle  points 
of  the    rafters.     If   the  weight    of 
the  roof-covering  and  snow  is  60  pounds  per  square 
foot,  find  the  apex  loads  AO,  AB,  and  BC. 


\J 


^2  MECHANICS-PROBLEMS. 

253.    Find  the  stresses  in  the 
king-post  truss  of  Fig.  31.     Dis- 
tance between  trusses  is  12  feet. 
^^'  ^'*  There  is  a  uniform  load  of   100 

pounds   per  square  foot  of  roof  surface  and    i  000 
pounds  at  the  foot  of  the  post. 

254.  A  king-post  truss  has  a  span  of  18  feet  and  a 
rise  of  9  feet.  Compute  the  stresses  due  to  a  load  of 
14  000  pounds  at  the  middle. 

255.  A  floor  beam  16  feet  long  and  carrying  a  uni- 
form load  of  200  pounds  per  linear  foot  is  trussed  by 
rods  that  are  \\  feet  below  middle  of  beam.  Con- 
sider a  joint  at  the  middle  and  find  stress  in  rod. 


MOMENTS 

The  principles  of  Work  can  be  used  to  solve  nearly 
all  problems  that  belong  to  the  subject  of  Mechanics, 
but  in  certain  classes  of  problems  shorter  methods 
are  possible.  In  the  following  problems  the  princi- 
ples of  Moments  can  be  used  to  advantage. 

Definition.  —  The  Moment  of  a  force  about  a  point  or  axis  is 
the  product  of  the  force  times  the  perpendicular  distance  from  the 
point  to  the  line  of  action  of  the  force  ;  or,  briefly;  Moment  is 
force  X  perpendicular. 

Clockwise  motion  will  be  taken  positive ;  the  opposite  direction, 
negative. 

In  beginning  the  solution  of  problems  always  state  which  point 
or  axis  the  moments  are  taken  about ;  thus,  "  Take  moments  about 
B,"  or  "  Moments  about  axis  B." 


FO/^CES  —  MOMENTS.  7  3 

256.  A  piece  of  shafting  10  feet  long,  and  weighing 
100  pounds,  rests  horizontally  on     ^  '^      ^  ^' 

two   horses  placed   at   its   ends,  ^f £__l_!^_^ 

A  pulley  weighing  75  pounds  is  ^'  J,^_,,\^ 

keyed    2  J    feet   from    one    end.  Fi«-  3a. 

How  many  pounds  will  a  man  have  to  lift  at  the  other 
end  to  just  raise  it.-* 

100  pounds,  the  weight  of  shaft,  acts  downward  at  the  middle 
point;  75  pounds,  the  weight  of  pulley,  acts  downward  at  D,  2 J  feet 
from  B.     Find  the  required  force  acting  upward. 

Take  moments  about  B, 

+  P  X  10  -  100  X  5  -  75  X  2^  =  o. 
.*.  P  =  67.5  pounds. 

257.  A  uniform  lever  is  18  inches  long,  and  each 
inch  in  length  weighs  i  ounce.  P'ind  the  place  of  the 
fulcrum  when  a  weight  of  27  ounces  at  one  end  of  the 
lever  balances  a  weight  of  9  ounces  at  the  other  end. 

258.  A  lever  16  feet  long  balances  about  a  point 
4  feet  from  one  end;  if  a  weight  of  120  pounds  be 
attached  to  the  other  end,  it  balances  about  a  point 
6  feet  from  that  end.     Find  the  weight  of  the  lever. 

259.  A  light  rod  of  length  3  yards  has  weights  of 

15  pounds  and  3  pounds  suspended  at  the  middle  and       \/ 
end  respectively  ;  it  balances  on  a  fulcrum.     Find  the 
position  of  the  fulcrum,  and  the  pressure  on  it. 

260.  A  stiff  pole  12  feet  long  sticks  out  horizon- 
tally from  a  vertical  wall.  It  would  break  if  a  weight 
of  28  pounds  were  hung  at  the  end.  How  far  out 
along  the  pole  may  a  boy  of  weight  1 1 2  pounds  ven- 
ture with  safety } 


y 


74  MECHANICS-PROBLEMS. 

261.  A  man  pulls  lOO  pounds  on  the  end  of  a  7-foot 
oar  that  has  2\  feet  inside  the  rowlock.  What  is  the 
pressure  on  the  rowlock,  and  resultant  pressure  caus- 
ing the  boat  to  move  ? 

262.  Find  the  propelling  force  on  an  eight-oared 
shell,  if  each  man  pulls  his  oar  with  a  force  of  56 
pounds,  and  the  length  of  the  oar  outside  the  row- 
lock is  three  times  the  length  inside. 

263.  A  light  bar,  5  feet  long,  has  weights  of  9 
pounds  and  5  pounds  suspended  from  its  ends,  and  10 
pounds  from  its  middle  point.    Where  will  it  balance  } 

^264.  A  weightless  lever  AB  of  the  first  order,  8 
feet  long,  with  its  fulcrum  2  feet  from  B,  has  a  weight 
of  5  pounds  hung  from  A,  and  one  of  17  pounds 
from  B.  From  what  point  must  a  weight  of  2.5 
pounds  be  hung  to  keep  the  lever  horizontal  1 

265.  A  weight  of  100  pounds  is  supported  by  a 
rope  which  passes  over  a  fixed  pulley  and  is  attached 
to  a  12-foot  lever  at  a  point  2  feet  from  the  fulcrum 
which  is  at  the  end.  What  weight  must  be  sus- 
pended at  the  other  end  to  keep  the  lever  horizontal .'' 

266.  Eight  sailors  raise  an  anchor,  of  weight  2  688 
pounds,  by  pulling  on  the  spokes  of  a  capstain  which 
has  a  radius  of  14  inches.  If  they  all  pull  at  equal 
distances  from  the  center  and  exert  a  force  of  56 
pounds  each,  what  is  the  distance  } 

267.  Is  there  any  reason  why  a  man  should  put 
his  shoulder  to  the  spoke  of  the  wheel  rather  than  to 
the  body  of  the  wagon  in  helping  it  up  hill  ? 


FORCES— MOMENTS.  75 

268.  A  rod  AB,  of  length  1 5  feet,  is  supported  by 
props  at  A  and  B  ;  a  weight  of  200  pounds  is  sus- 
pended from  the  rod  at  a  point  7  feet  from  A.  Find 
the  pressure  on  the  prop  at  A. 

269.  A  Hght  bar,  9  feet  long,  to  which  is  attached 
a  weight  of  150  pounds,  at  a  point  3  feet  from 
one  end,  is  borne  by  two  men.  Find  what  portion  of 
the  weight  is  borne  by  each  man,  when  the  bar  is 
horizontal. 

270.  A  light  rod,  16  inches  long,  rests  on  two  pegs 
9  inches  apart,  with  its  center  midway  between  them. 
The  greatest  weights,  which  can  be  suspended  sepa- 
rately from  the  two  ends  of  the  rod  without  disturb- 
ing the  equilibrium,  are  4  pounds  and  5  pounds  re- 
spectively. There  is  another  weight  fixed  to  the  rod. 
Find  that  weight  and  its  position. 

271.  A  light  rod  AB,  20  inches  long,  rests  upon 
two  pegs  whose  distance  apart  is  equal  to  half  the 
length  of  the  rod.  How  must  it  be  placed  so  that  the 
pressure  on  the  pegs  may  be  equal  when  weights  2W, 
3W,  are  suspended  from  A,  B,  respectively  ? 

272.  The  horizontal  roadway  of  a  bridge  is  30  feet 
long  and  its  weight,  6  tons,  may  be  supposed  to  act 
at  its  middle  point,  and  it  rests  on  similar  supports 
at  its  ends.  What  pressure  is  borne  by  each  of  the 
supports  when  a  carriage  weighing  2  tons  is  one-third 
of  the  way  across  the  bridge } 


u 


^6  MECHANICS-PROBLEMS. 

273.  "We  have  a  set  of  hay-scales,  and  some- 
times we  have  to  weigh  wagons  that  are  too  long  to 
go  on  them.  Can  we  get  the  correct  weight  by 
weighing  one  end  at  a  time  and  then  adding  the  two 
weights .?" 

274.  A  rod,  i8  inches  long,  can  turn  about  one  of 
its  ends,  and  a  weight  of  5  pounds  is  fixed  to  a  point 
6  inches  from  the  fixed  end.  Find  the  force  which 
must  be  applied  at  the  other  end  to  preserve  equilib- 
rium. 

275.  A  straight  uniform  lever  weighing  10  pounds 
rests  on  a  fulcrum  one-third  of  its  length  from  one 
end ;  it  is  loaded  with  a  weight  of  4  pounds  at 
that  end.  Find  what  vertical  force  must  act  at  the 
other  end  to  keep  the  lever  at  rest. 

276.  A  weight  ot  56  pounds  is  attached  to  one  end 
of  a  uniform  bar  which  is  ten  feet  long,  and  weighs 
20  pounds ;  the  fulcrum  is  2  feet  from  the  end  to 
which  the  weight  is  attached.  What  weight  must  be 
applied  at  the  other  end  to  balance  } 

277.  AB  is  a  horizontal  uniform  bar  \\  feet  long,  and 
F  a  point  in  it  10  inches  from  A.  Suppose  that  AB 
is  a  lever  turning  on  a  fulcrum  under  F,  and  carrying 
a  weight  of  40  pounds  at  B  ;  weight  of  lever,  4  pounds. 
If  it  is  kept  horizontal  by  a  fixed  pin  above  the  rod,  7 
inches  from  F  and  3  inches  from  A,  find  the  pressure 
on  the  fulcrum  and  on  the  fixed  pin. 


FORCES-MOMENTS. 


77 


278.  An  ununiform  rod,  i6  feet  long,  weighing  4 
pounds,  balances  about  a  point  4  feet  from  one  end. 
If,  2  feet  from  this  end,  a  weight  of  10  pounds  be 
hung,  what  weight  must  there  be  hung  from  the  other 
end  so  that  the  rod  may  balance  about  its  middle 
point .? 

279.  Six  men  are  to  carry  an  iron  rail  60  feet  long 
and  weighing  90  pounds  per  yard ;  each  man  sustains 
one-sixth  of  the  weight.  Two  men  are  to  lift  from 
one  end  and  the  other  four  by  means  of  a  cross-bar. 
Where  must  the  cross-bar  be  placed  ? 

280.  A  rod  2  feet  long,  with  a  weight  of  7  pounds 
at  its  middle  point,  is  placed  upon  two  nails,  A  and  B. 
AB  is  horizontal  and  7  inches  long.  Find  how  far 
the  ends  of  the  rod  must  extend  beyond  the  nails, 
if  the  difference  of  the  pressures  on  the  nails  be 
5  pounds. 

281.  A  davit  is  supported  by  a  foot- 
step A  and  a  collar  B,  placed  5  feet 
apart.  A  boat  weighing  two  tons  is 
about  to  be  lowered,  and  is  hanging 
4  feet  horizontally  from  vertical  through 
the  foot-step  and  collar.  Determine 
the  forces  which  must  be  acting  at  A 
and  B. 

282.  A  highway  bridge  of  span  50  feet,  breadth 
40  feet,  has  two  queen-post  trusses  of  depth  8  feet ; 
and  each  truss  is  divided  by  two  posts  into  three 
equal  parts.     The  bridge  is  designed  to  carry  a  load 


Fig.  33. 


1/ 


78 


ME  CHA  NICS-PROBLEMS. 


of   lOO  pounds  per  square  foot  of  floor  surface.     Find 
the  stresses  developed. 

Find  the  loads  for  each  truss  at  the  two  panel  points  C  and  D ; 
then,  by  the  methods  of  Moments,  find  the  reactions  R  and  R^  ob- 
serving, as  explained  for  problem  251,  that  at  each  end  half  a  panel 
of  the  load  goes  directly  on  the  abutment  and  does  not  affect  our 
computation  of  stresses  in  the  members  of  the  truss.  The  reactions 
thus  known  makes  it  possible  to  find  the  two  unknown  forces 
(stresses  in  the  members)  at  the  abutments.  Likewise  at  foot  of 
posts  three  forces  meet  in  a  point.  One  is  known,  —  the  stress  in 
post  which  is  equal  to  load  at  C  or  D,  —  and  the  other  two  can  be 
found  by  methods  of  three  forces  acting  at  a  point. 


C                 I 

i       50/«.        , 
^7 777-/777  T7:r'. •.:>.'.'.•.' 

3 

\ 

'777.7 

—  ^ 

R 

00 

K— 

50ft. 

— -»1 

W77f7 

~7;'777.'. 

VTr.rTZ/.  V7.  'TTT/,  777, 7777i 

R 

3()° 

\ 

/!. 

10,OOOW 

Pig.  34.  Fig-  35- 

283.  A  king-post  truss  of  20  feet  span,  as  shown 
in  Fig.  35,  has  a  uniform  load  of  10  X  200  pounds  on 
the  horizontal  member  and  10  000  pounds  at  the  foot 
of  the  post.     Determine  the  reactions  and  stresses. 

284.  A  5 -foot  water-pipe  is  carried  across  a  gully 
by  two  king-post  trusses  that  are  spaced  6  feet  apart. 
The  pipe  when  filled  with  water  makes  a  load  of  200 
pounds  per  square  foot.  Length  of  trusses  is  40 
feet ;  depth,  5  feet.     Find  the  stresses. 

285.  A  storehouse  has  queen-post  trusses  in  the 
top  story  ;  50  feet  span,  10  feet  depth,  lower  chord 
divided  into  3  equal  parts  ;  trusses  8  feet  apart,  and 
load  1 50  pounds  per  square  foot.     Find  the  stresses. 


FORCES— MOMENTS.  79 

286.  A  ladder  with  21  rungs  a  foot  apart  leans 
against  a  building  with  inclination  of  45°.  Find  the 
pressure  against  the  building  when  a  man  weighing 
1 50  pounds  stands  on  the  eleventh  rung. 

287.  Like  parallel  forces  of   10  and  20    units  act 
perpendicularly  to  AB  at  A  and  B  ;  a  force  of   15    . 
units  acts  from  A  to  B.     Find  the   resultant  of  the 
three  forces,  and  show  in  a  diagram  how  it  acts. 

288.  A  rod  is  acted  on  at  one  end  by  a  force  of  3 
downwards,  and  at  a  distance  of  two  feet  from  this 
end  by  a  force  of  5  upwards.  Where  must  a  force  of 
2  be  applied  to  keep  the  rod  at  rest  1 

289.  Three  parallel  forces  of  i  pound  each  act  on 
a  horizontal  bar.  The  right  hand  one  acts  vertically 
upwards,  the  two  others  vertically  downwards,  at  dis- 
tances 2  feet  and  3  feet  respectively,  from  the  first. 
Draw  their  resultant,  and  state  exactly  its  magnitude 
and  position. 

290.  A  rod  is  suspended  horizontally  on  two  points, 
A  and  B,  1 2  feet  apart ;    between  A  and    B  points 

C  and  D  are  taken,  such  that  AC  =  BD  =  3  feet ;  a  *^ 
weight  of  1 20  pounds  is  hung  at  C,  and  a  weight  of 
240  pounds  at  D  ;  the  weight  of  the  rod  is  neglected. 
Take  a  point  O,  midway  between  A  and  B,  and  find 
with  respect  to  O  the  algebraical  sum  of  the  moments 
of  the  forces  acting  on  the  rod  on  one  side  of  O. 

291.  A  horizontal  rod  without  weight,  6  feet  long, 
rests  on  two  supports  at  its  extremities  ;  a  weight  of 


80  MECHANICS-PROBLEMS. 

672  pounds  is  suspended  from  the  rod  at  a  distance  of 
2\  feet  from  one  end.  Find  the  reaction  at  each 
point  of  support.  If  one  support  could  bear  a  pres- 
sure of  only  1 1 2  pounds,  what  is  the  greatest  distance 
from  the  otlier  support  at  which  the  weight  could  be 
suspended  ? 

292.  Three  equal  parallel  forces  act  at  the  corners 
of  an  equilateral  triangle.  Find  the  point  of  applica- 
tion of  their  resultant.. 


293.  Find  the  center  of  the  three  parallel  forces  4 
pounds,  6,  and  8,  which  act  respectively  at  the  cor- 
ners of  an  equilateral  triangle. 

294.  P,  Q,  R,  are  parallel  forces  acting  in  the  same 
direction    at    the  angular   points    respectively  of  an 
equilateral  triangle  ABC.     If  P  =  2O  =  3R,  find  the        iy 
position  of  their  center ;  also  find  its  position  if  the 
direction  of  the  force  Q  is  reversed. 

295.  Show  that  if  two  forces  be  represented  in 
magnitude  and  direction  by  two  sides  of  a  triangle, 
taken  in  order,  the  sum  of  their  moments  about  every 
point  in  the  base  is  the  same. 

296.  Draw  a  square  whose  angular  points  in  order 

are  A,  B,   C,  D,  and  suppose  equal  forces  (P)  to  act         v 
from  D  to  A,  A  to  B,  and  B  to  C  respectively,  and  a 
fourth  force  (2P)  to  act  from  C  to  D.     Find  a  point 


FORCES— MOMENTS. 


8l 


such  that,  if  the   moments  of  the  forces  are  taken 
with  respect  to  it,  the  algebraic  sum  is  zero. 

297.    A  BCD   is  a  square,  the  length  of  each   side 

being  4  feet,  and  four  forces  act  as  follows  :  2  pounds 

from   D  to  A,  3  pounds  from  B  to  A, 

4  pounds  from  C  to   B,  and   5   pounds 

from  D  to  B.     Find  the  algebraical  sum 

of  the  moments  of  the  forces  about  C. 

a"" 

Fig.  36. 


The  forces 

act  as  in  the  figure. 

Draw 
Then, 

CM 

perpendicular  to  DB. 
CM  = 

DM. 

.-.  CD2=CM2  +  MD2 

=  2CM2. 

.-.  CM  = 

CD 
"^2 

.-.  CM  = 

4    = 

2.83  nearly 

.-.  Algebraical  sum  of  the  moments  about  C 

=  -  2  X  DC  +  3  X  CB  +  4  X  o  —  5  X  CM 
=  -2  X  4  +  3^X4  +0^5  (2.83) 
=  —  8  +  12  "^'14.15 
=  —  10.15  units. 

298.  ABCD  is  a  square,  and  AC  is  a  diagonal : 
forces  P,  Q,  R,  act  along  parallel  lines  at  B,  C,  D,  re- 
spectively, Q  acts  in  the  direction  A  to  C,  P  opposite 
direction,  and  R  in  opposite  direction.  Find,  and 
show  in  a  diagram,  the  position  of  the  center  when 
Q  =  5P  and  R  =  7P. 

299.  Draw  a  rectangle,  ABCD,  such  that  the  side 
AB  is  three-fourths  of  the  side  BC  ;  forces  of  3,  9, 
and  5  units  act  from  B  to  A,  B  to  C,  and  D  to  A  re- 
spectively.    Find  their   resultant  by  construction  or 


S2 


MECHANICS-rROBLEMS. 


otherwise,  and  show  in  your  diagram  exactly  how  it 
acts. 

300.  Prove  that,  if  parallel  forces  i,  2,  3,  4,  5,  6, 
are  situated  at  the  angles  of  a  regular  hexagon,  the 
distance  of  their  center  from  the  center  of  the  cir- 
cumscribing circle  is  two-sevenths  of  the  radius  of 
that  circle. 

301.  Six  forces,  represented  by  the  sides  of  a 
regular  hexagon  taken  in  order,  act  along  the  sides 
to  turn  the  hexagon  round  an  axis  perpendicular  to  its 
plane.  Show  that  the  moment  of  the  forces  is  the 
same  through  whatever  point  within  the  hexagon  the 
axis  passes. 

302.  A  triangular  table,  sides  8 
feet,  9  feet,  and  10  feet,  is  sup- 
ported by  legs  at  each  corner,  and 
350  pounds  is  placed  on  it  3  feet 
from  the  8-foot  side,  2  feet  from 
the    9-foot     side,    and     2.6     feet 

from  the     lo-foot  side.     What  are  the  pressures  on 

the  legs  } 

303.    A    triangular  shaped  platform  right-angled  at 

A,  with    side    AB    10     "  =^^ 

feet  long,  side  AC  40 

feet    long,     is     loaded 

with     freight     at      50 

pounds  per  square  foot  ^^s.  38. 

surface.     Find  the  load  carried  by  each  of  the  three 

corner-posts. 


10  u« 


FORCES—  MOMENTS.  8$ 

O,  the  center  of  gravity,  is  at  one-third  the  distance  from  the  mid- 
dle of  any  base  to  the  opposite  vertex.     Load  equals  lo  ooo  pounds. 

Take  moments  about  axis  AB  —  thus  find  load  carried  by  C. 
Then  take  moments  about  sides  AC  and  BC. 

304.  Four  vertical  forces,  5,  7,  10,  and  12  pounds, 
act  at  the  corners  of  a  square  of  20-inch  sides.  Find 
resuhant  and  its  point  of  application. 

Let  A  BCD  be  the  square, 

Resultant  =  5  +  7+  10+  12 
=  34  pounds. 
To  find  its  point  of  application  : 
Resultant  of  7  and  10  will  be  a  force 
of    17  pounds   acting   from    point  in    line 
CB  distant  /-  of  20  inches  from  B.     The 
resultant  of  5  and  12  will  be    17    pounds 
acting  at  a  point   in  line   AD  distant   -^^ 
of  20  inches  from    A.     The  resultant   of 
these  two  resultants  will  be  a  force  of  17 
-f   17  pounds,  34  pounds,  acting  at  a  point 

half  way  between  them,  and  at  a  perpendicular  distance  from  AB  of 
\  of  [jV  X  20  +  /y  X  20]  =  7yV  inches. 

305.  A  floor  20  X  30  feet  is  supported  mainly  by- 
four  posts,  one  at  each  corner.  There  is  a  load  of  20 
pounds  per  square  foot  uniformly  distributed,  and  at 
point  O,  5  feet  from  30-foot  side  and  7  feet  from  20- 
foot  side,  there  is  a  metal  planer  weighing  5  tons. 
Find  the  load  on  each  post. 

306.  Weights  5,  6,  9,  and  7  respectively,  are  hung 
from  the  corners  of  a  horizontal  square,  27  inches  in 
a  side.  Find,  by  taking  moments  about  two  adjacent 
edges  of  the  square,  the  point  where  a  single  force 
must  be  applied  to  balance  the  effect  of  the  forces  at 
the  corners. 


84  .    MECHANICS-PROBLEMS. 

307.  A  uniform  beam,  weighing  400  pounds,  is 
suspended  by  means  of  two  chains  fastened  one  at 
each  end  of  the  beam.  When  the  beam  is  at  rest  it 
is  found  that  the  chains  make  angles  of  100°  and  115° 
with  the  beam.     Find  the  tensions  in  the  chains. 

308.  A  force  of  50  pounds  acts  eastward  and  a 
force  of  50  pounds  acts  westward.  Will  there  be 
motion  ? 

That  depends,  as  will  easily  be  seen,  upon  the  position  of  the 
forces.  If  they  act  on  the  two  ends  of  a  rope  there  will  be  no  mo- 
tion. If  they  act  one  on  the  northerly  part  of  a  brake  wheel  and 
one  on  the  southerly  part  there  will  be  motion,  —  that  of  rotation. 

Such  forces  produce  a 

Couple  :  two  equal,  opposite,  parallel  forces  not  acting  in  the 
same  straight  line. 

The  tendency  to  motion  by  couples  is  not  of  translation  but  of 
rotation.     The  measure  of  this  tendency  is,  — 

Moment  of  a  couple  equals  the  product  of  one  of  the  two  forces 
X  perpendicular  distance  between  them. 

What  is  the  resultant  of  a  couple  of  moment  15, 
and  a  force  3  ? 

309.  A  brakeman  sets  up  a  brake  on  a  freight 
car  by  pulling  50  pounds  with  one  hand  and  pushing 
50  pounds  with  the  other  ;  his  forces  act  tangentially 
to  the  brake  wheel,  the  diameter  of  which  is  \\  feet. 
Another  time  he  produces  the  same  brake  resistance 
by  using  a  lever  in  hand  wheel  and  pulling  25  pounds. 
How  far  from  handwheel  must  his  hands  be  placed  t 

310.  When  are  couples  said  to  be  like  and  when 
unUke  1     When  will  two  unlike  couples  balance  each 


FORCES  —  MOMEXTS.  8  5 

Other?  (i)  If  a  system  of  forces  is  represented  in 
magnitude  and  position  by  the  sides  of  a  plane  poly- 
gon taken  in  order,  show  that  the  system  must  be 
equivalent  to  a  couple.  (2)  If  the  sides  of  a  parallelo- 
gram taken  in  order  represent  a  system  of  forces  act- 
ing upon  a  body,  express  the  moment  of  the  couple  to 
which  the  system  of  forces  is  equivalent. 

311.  Show  that  a  force  and  a  couple  in  one  plane 
may  be  reduced  to  a  single  force.  Given  in  position 
a  force  of  10  pounds,  and  a  couple  consisting  of  two 
forces  of  4  pounds  each,  at  a  distance  of  2  inches, 
acting  with  the  hands  of  a  clock,  draw  the  equivalent 
single  force. 

312.  The  length  of  the  side  of  a 
square  ABCD  is  12  inches.  Along 
the  sides  AB  and  CD  forces  of  10 
pounds  act,  and  along  AD,  CB  forces 
of  20  pounds.  Find  the  moment  of 
the  equivalent  couple. 

Moments  about  D, 

—  12  X  10+12  X2o  =  moment  of  equivalent-couple 
12  X  10  =  moment  of  equivalent-couple 

313.  Forces  P  and  O  act  at  A,  and  are  completely 
represented  by  AB  and  AC,  sides  of  a  triangle 
ABC.  Find  a  third  force  R  such  that  the  three 
forces  together  may  be  equivalent  to  a  couple  whose 
moment  is  represented  by  half  the  area  of  the  triangle. 

314.  A  tradesman  has  a  balance  with  arms  of  un- 
equal length,  but  tries  to  be  fair  by  weighing  his  ma- 


S6  MECHANICS-PROBLEMS. 

terial  first  from  one  scale  pan,  then  from   the  other. 
Show  that  he  will  defraud  himself. 

315.  A  tradesman  uses  a  balance  with  arms  in 
ratio  of  5  to  6  ;  he  weighs  out  from  alternate  pans 
what  appears  to  be  30  pounds.  How  much  does  he 
gain  or  lose } 

316.  The  beam  of  a  balance  is  6  feet  long,  and  it 
appears  correct  when  empty ;  a  certain  body  placed 
in  one  scale  weighs  1 20  pounds,  when  placed  in  the 
other,  121  pounds.  Show  that  the  fulcrum  must  be 
distant  about  J3  of  an  inch  from  the  center  of  the 
beam. 

317.  The  weight  of  a  steelyard  is  1 2  pounds,  its 
movable  weight  is  3  pounds.  Find  the  distance 
between  successive  pound  graduations,  if  the  length 
of  the  short  arm  is  3  inches. 

318.  A  weight  of  247  pounds  is  attached  to  one 
end  of  a  horizontal  straight  lever,  which  is  22  inches 
long,  and  may  be  regarded  as  having  no  weight ; 
the  force  is  applied  at  the  other  end,  and  makes  an 
angle  of  27°  with  the  lever;  the  fulcrum  is  3  inches 
from  the  weight.  F'ind  the  magnitude  of  the  force 
when  it  just  balances  the  weight. 

319.  A  uniform  beam  rests  at  a 
given  inclination,  Q,  with  one  end 
against  a  smooth  vertical  wall,  and 
the  other  end  on  smooth  horizontal 
ground  :  it  is  held  from  slipping  by 
a  string  extending  horizontally  from 


FORCES  —  MOMENTS.  8  / 

the  foot  of  the  beam  to  the  foot  of  the  wall.  Find 
the  tension  in  the  string  and  the  pressures  at  the 
ground  and  wall. 

AB  is  the  beam,  AC  the  wall,  BC  the  string,  W  the  weight  of 
the  beam  acting  at  its  middle  point  G. 

There  are  three  forces  supporting  the  beam :  vertical  reaction  P, 
horizontal  reaction  R,  and  tension  in  the  string  F. 

T?.ke  moments  about  B,  the  point  of  intersection  of  two  of  the 

forces  —  their  lever  arms  would  be  zero. 

BC 
Rx  AC  =  W  X 

2 

Substitute  for  AC  its  value  BC  X  tan  d,  then 

W 


(.)R  = 


2  tan 


but  R  must  equal  F,  both  being  horizontal  resisting  forces  that  main- 
tain equilibrium;  likewise  P  and  W  must  be  equal. 

...  (2)  F  =      ^^    ,  and 
^  '  2  tan  e 

(3)  P  =  W 

320.  A  uniform  beam  rests  with  a  smooth  end 
against  the  junction  of  the  horizontal  ground  and  a 
vertical  wall ;  it  is  supported  by  a  string  fastened  to 
the  other  end  of  the  beam  and  to  a  staple  in  the  ver- 
tical wall.  Find  the  tension  of  the  string,  and  show 
that  it  will  be  half  the  weight  of  the  beam  if  the 
length  of  the  string  be  equal  to  the  height  of  the 
staple  above  the  ground. 

321.  A  uniform  rod  8  feet  long,  weighing  i8 
pounds,  is  fastened  at  one  end  to  a  vertical  wall  by  a 
smooth  hinge,  and  is  free  to  move  in  a  vertical  plane 
perpendicular  to  the  wall.  It  is  kept  horizontal  by  a 
string  lo  feet  long,  attached  to  its  free  end  and  to  a 


88  MECHANICS-PROBLEMS. 

point  in  the  wall.      Find  the  tension  in  the  string,  and 
the  pressure  on  the  hinge. 

322.  A  uniform  beam,  1 2  feet  in  length,  rests  with 
one  end  against  the  base  of  a  wall  which  is  20  feet 
high.  If  the  beam  be  held  by  a  rope  13  feet  long, 
attached  to  the  top  of  the  beam  and  to  the  summit  of 
the  wall,  find  the  tension  of  the  rope,  neglecting  its 
weight,  and  assuming  the  weight  of  the  beam  to  be 
100  pounds. 

323.  ABC  is  a  rigid  equilateral  triangle,  weight  not 
considered;  the  vertex  B  is  fastened  by  a  hinge  to  a 
wall,  while  the  vertex  C  rests  against  the  wall  under 
B.  If  a  given  weight  is  hung  from  A,  find  the  reac- 
tions at  B  and  C.  What  are  the  magnitudes  and 
directions  of  the  forces  exerted  by  the  weight  on  the 
wall  at  B  and  C  1 

324.  A  beam  AB  rests  on  the  smooth  ground  at 
A  and  on  a  smooth  inclined  plane  at  B  ;  a  string  is 
fastened  at  B  and,  passing  over  a  smooth  peg  at  the 
top  of  the  plane,  supports  a  weight  P.  If  W  is  the 
weight  of  the  beam,  and  a  the  inclination  of  the  plane, 
find  P  and  the  reactions  on  the  rod. 

Draw  the  figure. 

The  weight  W  acts  at  the  middle  point  C,  The  reaction  of  the 
ground  at  A  is  R,  upwards. 

The  reaction  of  the  plane  at  B  is  Ri,  perpendicular  to  the  plane. 

Let  the  angle  BAD  =  Q. 

The  tension  of  the  string  at  B  =  tension  of  the  string  throughout 

:^   P 

There  are  four  forces  acting  on  the  beam,  W,  R,  Ri,  P. 
Resolve  vertically  and  horizontally. 


FORCES — MOMENTS.  89 

325.  A  pole  12  feet  long,  weighing  25  pounds, 
rests  with  one  end  against  the  foot  of  a  wall,  and 
from  a  point  2  feet  from  the  other  end  a  cord  runs 
horizontally  to  a  point  in  the  wall  8  feet  from  the 
ground.  Find  the  tension  of  the  cord  and  the  pres- 
sure of  the  lower  end  of  the  pole. 

326.  A  light  smooth  stick  3  feet  long  is  loaded  at 
one  end  with  8  ounces  of  lead  ;  the  other  end  rests 
against  a  smooth  vertical  wall,  and  across  a  nail  which 
is  I  foot  from  the  wall.  Find  the  position  of  equi- 
librium and  the  pressure  on  the  nail  and  on  the  wall. 

327.  A  trapezoidal  wall  has  a  vertical  back  and  a 
sloping  front  face  ;  width  of  base,  i  o  feet ;  width  of 
top,  7  feet ;  height,  30  feet.  What  horizontal  force 
must  be  applied  at  a  point  20  feet  from  the  top  in 
order  to  overturn  it }  Thickness  of  wall,  i  foot ; 
weight  of  masonry  in  Avail,  1 30  pounds  per  cubic  foot. 

328.  Six  men  using  a  rope  50  feet  long  were  just 
able  to  pull  over  a  chimney  7$  feet  high.  How  far 
up  from  the  bottom  of  the  chimney  was  it  advisable 
to  attach  the  rope  } 

329.  If  1 50  000  pounds  is  the  thrust  along  the 
connecting  rod  of  the  engine,  in  example  %6y  2 J  feet 
the  crank  radius,  and  the  connecting-rod  is  inclined 
to  the  crank  axis  at  150°,  show  that  the  moment  of 
the  thrust  about  the  crank-pin  is  one-half  the  greatest 
possible  moment. 

330.  A  trap-door  of  uniform  thickness,  5  feet  long 
and  3  feet  wide,  and  weighing  5  hundred  weight,  is 


90  MECHANICS-PROBLEMS. 

held  open  at  angle  of  35°  with  the  horizontal  by 
means  of  a  chain.  One  end  of  chain  is  hooked  at 
middle  of  top  edge  of  door,  and  the  other  is  fastened 
at  wall  4  feet  above  hinges.  Find  the  force  in  the 
chain  and  the  force  at  each  hinge. 

331.    The  sketch  represents  a  coal  wagon  weighing 

with  its  load  4|  tons.     How 
many    pounds    applied   at 
P  by  usual  methods  of  hand 
-^B  power   will    just    lift    the 
wagon  when   in    the   posi- 
tion shown  in  the  sketch  } 
AE   is  a  rod   in  tension.     CD   is  a  connecting-bar. 
Divide  the  problem  into  three  parts : 
(a)    Draw  the  forces  acting. 

{b )    Find  horizontal  distance  from  C  to  the  verti- 
cal through  the  center  of  gravity. 

( e)    Find  force  to  apply  at  C  parallel  to  P ;  then 
find  P. 


CENTER     OF     GRAVITY 

332.  A  rod  of  uniform  section  and  density,  weigh- 
ing 3  pounds,  rests  on  two  points,  one  under  each 
end  ;  a  movable  weight  of  4  pounds  is  placed  on 
the  rod.  Where  must  it  be  placed  so  that  one  of  the 
points  may  sustain  a  pressure  of  3  pounds,  and  the 
other  a  pressure  of  4  pounds  ? 


FORCES— CENTER    OF  GRAV/TV.  9 1 


333.  Two  rods  of  uniform  density 
weighing  2  pounds  and  3  pounds  re- 
spectively are  put  together  so  that  the 
3-pound  one  stands  on  the  middle  of 
the  other.     Find  the  center  of  gravity  of  a- 


the  whole.  ""=•  ''■ 

Take  moments  about  AB, 

+  3  X  J  /  -  5  X  a:  =  o 

334.  A  thin  plate  of  metal  is  in  the  shape  of  a 
square  and  equilateral  triangle,  having  one 
side  common  ;  the  side  of  the  square  is  1 2 
inches  long.  Find  the  center  of  gravity  of 
the  plate. 

B  Let  Gi  be  the  center  of  gravity  of  the  triangle,  Gj 

of  the  square,  G  of  the  whole  plate. 
From  symmetry  EGj  GG^O  will  be  a  straight  line  bisecting  the 
plate,  and 

OG.,  =  6  inches 
OGi  =15.5  inches 
Let  7^  =  weight  of  metal  per  square  inch 

Area  of  triangle  =  ^  x  12  x^iz'^  —  6'^ 
=  62.4  square  inches 
Weight  =  62.4  pounds  X  w  pounds 
Area  of  square  =  144  square  inches 
Weight  =  144  X  w  pounds 
Take  moments  about  the  axis  AB, 
Weight  of  triangle  X  OGi  +  weight  of  square  XOG2— total 
weight  X  OG  =  o 
62.4 wx  i5.5-|-i44Z£/x6  —  (62.4«/-|-i447£/)  X  OG  =  o 
.-.  OG  =  8.86  inches. 


92 


MECHANICS-PROBLEMS, 


335.  A  bridge  member  has  two  web  plates  i8 
X  I  inches,  top  plate  21  x  g,  top  angles  3x3  and 
I  inches  thick,  bottom  angles  4x3  and  y^.  inches 
thick.  Find  "  eccentricity  "  —  the  distance  from  AB, 
the  neutral  axis  through  the  center  of  gravity  to  C, 
the  middle  of  the  section. 


n 


B 


L 

Fig.  45.  Fig.  46. 

336.  Web  plate  of  Fig.  46  is  10  x  \  inches,  top 
plate  12  X  J,  two  angles  4  x  3  X  |.  Find  "eccen- 
tricity."     (Given  in  Osborn's  Tables  (1905)  page  24.) 

337.  Fig.  47  shows  a  cross-section  of  the  top  chord 
of  one  of  the  main  trusses  in  the  Portage  Canal 
Draw-Bridge  at  Houghton,  Mich.  See  Engineering 
News  of  June  15,  1905.  In  computing  the  strength 
of  this  built-up  member,  it  is  required  to  find  the 
position  of  the  axis  AB  that  passes  through  the  center 
of  gravity  of  the  section. 


I  3ix-3iy| 


A 


■fSSSS.SSSVt- 


s.\ssssss  ess 


\- 


Fig.  47- 


-^f 


3iX3M 
angle 


l^» 


kK\N\\-.\^ 


Fig.  48. 


FORCES— CENTER  OF  GRAVITY,  93 

338.  The  strength  of  steel  rails  is  usually  com- 
puted by  embodying,  among  other  factors,  the  distance 
from  neutral  axis,  which  passes  through  the  center  of 
gravity,  to  the  extreme  fibres  of  the  section.  A 
100-pound  rail,  of  the  Lorain  Steel  Company,  has  a 
section  shown  in  Fig.  48.  Draw  the  section  carefully 
to  full  scale  on  bristol  board ;  then  cut  it  out  and 
locate  its  center  of  gravity  by  balancing  on  a  knife 
edge.  What  is  the  distance  from  center  of  gravity 
to  extreme  fibres  ? 

339.  ABC  is  a  triangle  with  a  right  angle  at  A. 
AB  =  3  inches ;  AC  =  4  inches ;  weights  of  2 
ounces,  3  and  4,  are  placed  at  A,  B,  and  C.  Find 
the  position  of  their  center  of  gravity. 

340.  A  uniform  triangle  ABC  of  weight  W,  and 
lying  on  a  horizontal  table,  is  just  raised  by  a  vertical 
force  applied  at  A.  P^ind  the  magnitude  of  this  force, 
and  that  of  the  resultant  pressure  between  the  base 
BC  and  the  table. 

341.  A  uniform  circular  disk  has  a  circular  hole 
punched  out  of  it,  extending  from  the  circumference 
half  way  to  the  center.  Find  the  center  of  gravity 
of  the  remainder. 

342.  A  box,  including  its  cover,  is  made  of  six 
equal  square  boards ;  where  is  its  center  of  gravity 
when  its  lid  is  turned  back  through  an  angle  of  180°  t        i 


94 


MECHANICS-PROBLEMS, 


343.    ABCD  is  a 

thin       rectangular 
plate    weighing    50  e; 
pounds,    AB    is    10  nj 
feet,  EC  2  feet ;  the  ^  i 
plate    is    suspended 
by  the  middle  point 
of   its    upper    edge 
AB,    and    then,    of  ^'^- ^^' 

course,  AB  is  horizontal,  but  if  a  weight  of  5  pounds 
is  placed  at  A,  AB  will  become  inclined  to  the  hori- 
zon. Show  how  to  find  the  angle  of  inclination 
either  by  calculation  or  by  construction. 


344.  A  circular  disk,  8  inches  in  diameter,  has  a 
hole  2  inches  in  diameter  punched  out  of  it,  the  center 
of  the  hole  being  3  inches  from  the  circumference 
of  the  disk.  Find  the  center  of  gravity  of  the  remain- 
ing portion. 


345.  Find  the  centers 
of  area  of  the  above  sec- 
tions of  uniform  plate 
metals. 


<  5  in.    > 

<  3  >  o 


Fig.  50. 


346.  Into  a  hollow  cylindrical  vessel  1 1  inches 
high  and  weighing  10  pounds,  the  center  of  gravity 
of  which  is  5  inches  from  the  base,  a  uniform 
solid  cylinder  6  inches  long  and  weighing  20  pounds 
is  just  fitted.     Find  the  common  center  of  gravity. 


K 

-^Gi 

i 

.4Go 

o 

-JC 

FORCES— CENTER    OF  GRAVITY.  95 

Gi  center  of  gravity  of  hollow  cylinder 
G2  center  of  gravity  of  solid  cylinder. 
Moments  about  AB, 

+  10X5  +  20X3  —  3oXJc=o 

+  50  +  60  —  30  AT  =  O 
30^:=    I  10 

X  =  3§  inches.  ^*K-  51. 

347.  Give  examples  of  stable  and  unstable  equilib- 
rium. A  cone  and  a  hemisphere  of  the  same  material 
are  cemented  together  at  the  common  circular  base. 
If  they  are  on  a  horizontal  plane,  and  the  hemisphere 
in  contact  with  the  plane,  find  the  height  of  the  cone 
in  order  that  the  equilibrium  may  be  neutral.  (The 
center  of  gravity  of  a  hemisphere  divides  a  radius  in  the 
ratio  of  3  to  5.) 

348.  A  thread  9  feet  long  has  its  ends  fastened  to 
the  ends  of  a  rod  6  feet  long ;  the  rod  is  supported 
in  such  a  manner  as  to  be  capable  of  turning  freely 
round  a  point  2  feet  from  one  end  ;  a  v^eight  is  placed 
on  the  thread,  like  a  bead  on  a  string.  Find  the 
position  in  which  the  rod  will  come  to  rest,  it  being 
supposed  that  the  rod  is  without  weight,  and  that  there 
is  no  friction  between  the  weight  and  the  thread. 

349.  A  circular  disk  weighs  9  ounces ;  a  thin 
straight  wire  as  long  as  the  radius  of  the  circle  weighs 
7  ounces  ;  if  the  wire  is  placed  on  the  disk  so  as  to  be 
a  chord  of  the  circle,  the  center  of  gravity  of  the 
whole  will  be  at  a  distance  from  the  center  of  the 
circle  equal  to  some  fractional  part  of  the  radius. 
Find  that  fraction  by  construction  or  calculation. 


y 


96  MECHANICS-PROBLEMS, 

350.  A  cone  and  a  hemisphere  are  on  the  same 
base.  What  height  must  the  cone  be  in  order  that  the 
center  of  gravity  of  the  whole  solid  shall  be  at  the 
center  of  the  common  base  ? 

r  =  radius  common  base. 
h  =  height  of  cone. 


FRICTION 

The  coefficients  of  friction  for  various  pairs  of  sub- 
stances have  been  found  experimentally  by  Morin ; 
these  results  however  can  be  used  only  for  approxi- 
mate computation ;  actual  trial  should  be  made  for 
specific  cases.     Average  values  are: 

Stone  on  stone 0.40  to  0.65 

Wood  on  wood 0.25  to  0.40 

Metal  on  metal,  dry  .     .     ...     .     .     0.15  to  0.30 

well  oiled       .     .     .     0.0 1  to  o.io 

1.  Friction  is  proportional  to  normal  reaction,  R. 

2.  Is  independent  of  area  of  contact. 

3.  Is  dependent  very  much  on  the  roughness  of  surfaces. 

351.  Define  "  coefficient "  and  *' angle  of  friction," 
and  "resultant  reaction." 

R.  352.  A  weight  of  56  pounds  is  moved 

F^-^-^sib».   along  a  horizontal  table  by  a  force  of 
I        b6iis.        I  8  pounds.     How  much  is  the  coefficient 
Fig.  52.        of  friction  .? 

The  pull  of  8  pounds  is  required  to  overcome  friction,  and  is 
equal  to  the  friction. 

Friction  =  coefficient  x  Reaction  (perpendicular  to  plane  of  table. 


FORCES— FRICTION.  97 

F  =  /M  X  R 

=  /^i  X  56  pounds 
8  =  ya  X  56 

=  f 

353.  A  5  X  8-foot  vertical  gate  has  a  head  of  water 
against  its  center  equal  to  10  feet,  or  4 J  pounds  per 
square  inch.  The  coefficient  of  friction  being  0.40, 
what  force  is  required  in  raising  it  to  overome  the 
friction  1 

354.  A  horizontal  pull  of  50  pounds  is  required  to 
slide  a  trunk  along  the  floor.  The  coefficient  of  fric- 
tion is  0.20,  and  trunk  when  empty  weighs  75  pounds. 
How  many  pounds  of  goods  does  it  contain  } 

355.  A  block  of  stone  is  dragged  along  the  ground 
by  a  horse  exerting  a  force  of  224  pounds.    If  /x  =  0.6,   / 
what  is  the  weight  of  the  block } 

356.  A  weight  of  500  pounds  is  placed  on  a  table, 
and  can  hardly  be  slid  by  a  horizontal  pull  of  155 
pounds.  Find  the  coefficient  of  friction,  and  the 
number  of  degrees  in  the  angle  of  friction  by  measur- 
ing from  a  drawing  made  to  a  scale. 

357.  A  stone  just  slides  down  a  hill  of  inclination 
30°.     What  is  the  coefficient  of  friction  t  ^ 

358.  A  block  rests  on  a  plane  which  is  tilted  till 
the  block  commences  to  slide.  The  inclination  is 
found  to  be  8.4  inches  at  starting,  and  afterwards  6.3 
inches  on  a  horizontal  length  of  2  feet.     Find  the  co- 


98  MECHANICS-PROBLEMS. 

efficient  of  friction  when  the  block  starts  to  slide,  and 
after  it  has  started. 

359.  A  horse  draws  a  load  weighing  2  000  pounds 
up  a  grade  of  i  in  20 ;  the  resistance  on  the  level  is 
100  pounds  per  ton.  Find  the  pull  on  the  traces 
when  they  are  parallel  with  the  incline. 

360.  How  much  work  has  a  man,  weighing  224 
pounds,  done  in  walking  twenty  miles  up  a  slope  of  i 
vertical  to  40  horizontal  1  What  force  could  drag  a 
dead  load  of  the  same  weight  up  the  same  hill  {a)  if 
friction  be  negligible,  {b)  if  friction  be  \  of  the 
weight } 

361.  Three  artillerymen  drag  a  gun  weighing 
I  700  pounds  up  a  hill  rising  2 
vertically  in  17  horizontally.  Sup- 
pose the  resistance  to  the  wheels 
going   up  the  hill    be    16    pounds 

per  hundred  weight,  what  pull  parallel  to  the  hill  must 
each  exert  to  move  it } 

When  the  gun  is  about  to  move  forward  the  pull  P  will  be  acting 
up  the  plane,  and  parallel  to  it ;  the  friction  F  down  the  plane,  hold- 
ing back;  the  force  R  perpendicular  to  inclined  plane,  partly  sup- 
porting the  gun,  and  W  the  weight  of  the  gun  acting  vertically  down- 
ward. "Weight  of  gun  is  given —  i  700  pounds.  Resolve  into  com- 
ponents perpendicular  and  parallel  to  the  plane.  The  perpendicular 
component  will  be  the  supporting  force  of  the  plane  —  its  reaction 
R ;  the  parallel  component  will  be  the  part  of  the  pull  P  required  by 
weight  of  the  gun. 

362.  Find  the  force  which,  acting  in  a  given  direc- 
tion, will  just  support  a  body  of  given  weight  on  a 


FORCES  —  FRIC  TION.  99 

rough  inclined  plane.  The  height  is  to  the  base  of 
the  plane  as  3  to  4,  and  it  is  found  that  the  body  is 
just  supported  on  it  by  a  horizontal  force  equal  to  half 
the  weight  of  the  body.  Find  the  coefficient  of  fric- 
tion between  the  body  and  the  plane. 

363.  The  table  of  a  small  planing-machine  which 
weighs  1 1 2  pounds  makes  six  single  strokes  of  4^ 
feet  each  per  minute.  The  coefficient  of  friction  be- 
tween the  sliding  surfaces  is  .07.  What  is  the  work 
in  foot-pounds  per  minute  performed  in  moving  the 
table  ? 

364.  A  rectangular  block  ABCD  whose  height  is 
double  its  base,  stands  with  its  base  AD  on  a  rough 
floor,  coefficient  of  friction  J.  If  it  be  pulled  by  a 
horizontal  force  at  C  till  motion  ensues,  determine 
whether  it  will  slip  on  the  floor,  or  begin  to  turn  over 
round  D. 

365.  A  cubical  block  rests  on  a  rough  plank  with 
its  edges  parallel  to  the  edges  of  the  plank.  If,  as 
the  plank  is  gradually  raised,  the  block  turns  over  on 
it  before  slipping,  how  much  at  least  must  be  the 
coefficient  of  friction  t 

366.  A  weight  of  5  pounds  can  just  be  supported 
on  a  rough  inclined  plane  by  a  weight  of  2  pounds,  or 
can  just  support  a  weight  of  4  pounds  suspended  by 
a  string  passing  over  a  smooth  pulley  at  the  vertex. 
Find  the  coefficient  of  friction,  and  the  inclination  of 
the  plane. 


1 00  MECHA  NICS-FROBLEMS, 

367.  Find  the  least  force  that  will  drag  a  box 
weighing  200  pounds  along  a  concrete  floor,  the  co- 
efficient of  friction  being  0.50. 

The  required  force  will  of  course  not  act  horizontally,  but  instead 
in  some  direction  as  P.     To  find  the  angle  b : 
Resolve  vertically 

—  P  sin  (^  —  R  +,  200  =  o 
Resolve  horizontally 

+  P  cos  b  —  /otR=  o 
From  these  two  equations 

p_  200M  _ 

ju.  sin  <^  +  cos  b 

When  will  P  be  as  small  as  possible  ?  When  m  sin  b  +  cos  b  is 
as  large  as  possible.  The  student  not  familiar  with  the  calculus  can 
find  by  trial  that  the  maximum  value  of  denominator,  or  least  value 
of  the  pull  P,  will  occur  when  b  =  tan— ^  /ix,  that  is,  the  angle  whose 
tangent  is  m-. 

By  the  method  of  calculus, 

M  sin  b  +  cos  b  =  x 

Differentiate,  noting  that  m  is  a  constant ;  and,  to  find  a  critical 
value,  which  in  this  case  will  be  a  maximum  value,  place  the  first  dif- 
ferential equal  to  zero. 

dx  1         •     / 

— -  =  M  COS  b  —  s\n  b  =^  o 

db 

M  =  tan  b 

b  =  tan  ~^  ft. 

=26^34' 

p  _        200  X  1 

I  X  .447  +  .894 

=  89  pounds. 

368.  By  experiment  it  was  found  that  a  box  of  sand 
weighing  204  pounds  required  a  least  pull  of  iio 
pounds  (at  angle  a)  to  move  it  on  a  concrete  floor. 
What  was  the  value  of  /x  .?       . 


FORCES—  FRICTION. 


lOI 


369.  The  roughness  of  a  plane  of 
inclination  30°  is  such  that  a  body  of 
weight  500  pounds  can  just  rest  on 
it.  What  is  the  least  force  required 
to  draw  the  body  up  the  plane  ? 


Fig.  55. 
As  in  problem  367  a  will  equal  the  angle  of  friction,  or  tan  '  /i. 


370.  A  sled  of  total  weight  3  tons  is  to  be  drawn 
up  a  grade  of  i  vertical  to  8  horizontal.  The  coeffi- 
cient of  friction  between  the  sled  shoes  and  the  snow 
is  o.  10.  What  angle  should  the  traces  make  with  the 
horizontal }     What  pull  will  the  horse  exert } 


The  problems  that  pertain  to  the  wedge  can  be  solved  by  the  same 
methods  that  have  been  used  for  the  inclined  plane.  The  essential 
principles  are  :  Show  the  conditions  by  a  sketch,  indicating  carefully 
the  position  and  direction  of  all  forces  ;  then,  (i)  resolve  parallel  to 
plane.  (2)   resolve  perpendicular  to  plane.     Thus  for  problem  371 


Fig.  56  shows  the  conditions  for  one  form  of  wedge.  Fig.  57  for 
another. 

Observe  the  directions  of  R  and  W.  As  can  be  seen  in  Fig.  56, 
\  P  should  decrease  the  value  of  R,  therefore  R  must  act  in  the 
direction  indicated.  In  previous  problems  the  weight  has  moved  on 
the  inclined  plane  ;  here  the  plane  moves. 

Resolve  ||  to  top  plane  (Fig.  56), 

—  0.20  R  +  300  cos  3°  35'  —  W  X  sin  3°  35'  =  o. 


102  MECHANICS-PROBLEMS. 

Resolve  ±  to  plane, 

+  R-300  sin  3°  35'  -  W  X  cos  3°  35'  =  o. 

Solve  these  two  equations  for  W. 

To  find  the  pull  necessary  to  withdraw  the  wedge,  sketch  another 
figure  showing  /xR  and  5  P  in  their  new  positions.  Then  solve  as 
indicated  above. 

371.  A  cotter,  or  wedge,  having  a  taper*  of  i  in  8, 
is  driven  into  a  cottered  joint  with  an  estimated  pres- 
sure of  600  pounds.  Taking  the  coefficient  of  friction 
between  the  two  surfaces  as  0.2,  find  the  force  which 
the  wedge  exerts  at  the  joint  perpendicular  to  the 
pressure  of  600  pounds  ;  also  find  the  pull  necessary 
to  withdraw  the  wedge. 

372.  A  floor-column  with  its  load  of  5  tons  is  to  be 
lifted  by  two  wedges  driven  towards  each  other. 
Thickness  of  each  wedge  is  2  inches,  length,  12 
inches;  coefficient  of  friction,  0.15.  Find  the  force 
that  must  be  equivalent  to  P  in  order  to  drive  the 
wedge. 

373.  A  casting  of  weight  5  000  pounds  is  to  be 
lifted  by  an  iron  wedge  that  is  forced  ahead  by  a 
screw  and  mechanism  that  can  give  an  equivalent 
force  of  3  000  pounds.  If  a  12-inch  wedge  is  used, 
what  should  be  its  thickness  } 

374.  A  rough  wedge  has  been  inserted  into  a  block 
and  is  only  acted  on  by  the  reactions.  If  it  is  on  the 
point  of  slipping  out,  and  the  coefficient  of  friction 

is  — p ,  what  is  the  angle  of  the  wedge  .? 
V3 


FORCES—  FRICTION.  1 03 

375.  A  steel  wedge  1 2  inches  long,  2  inches  thick, 
tapering  on  both  sides  to  o,  is  used  to  wedge  up  a 
pump  plunger  weighing  3  000  pounds  by  means  of  a 
maul  weighing  5  pounds.  The  coefficient  of  friction 
is  0.15  and  the  striking  velocity  of  the  maul  is  25  feet 
per  second.    How  far  will  each  blow  drive  the  wedge  ? 

376.  A  wheel  of  weight  W  rests  between  two 
planes,  each  inclined  to  the  vertical  at  angle  a ;  the 
plane  of  the  wheel  is  perpendicular  to  the  line  of  in- 
tersection of  the  two  planes,  which  is  itself  horizontal. 
If  ft  be  the  coefficient  of  friction,  find  the  least  couple 
necessary  to  turn  the  wheel. 

377.  A  ladder  inclined  at  an  angle  of  60°  to  the 
horizon  rests  with  one  end  on  rough  pavement,  and 
the  other  end  against  a  smooth  vertical  wall ;  the 
ladder  begins  to  slide  down  when  a  weight  is  put  at 
its  middle  point.     Show  that  the  coefficient  of  friction 

is — ^  . 
6 

When  the  ladder  begins  to  slide  down,  the 
limiting  friction  would  be  /xR. 
Resolve  vertically, 

W  =  R 

Resolve  horizontally, 

R'  =mR 

Take  moments  about    B,   and   then   solve 
for  li.. 

If  the  wall  should  be  rough  there  would  be 
acting  at  B  an  upward  force  of  /x'R'  that  would  have  to  be  embodied 
in  the  above  equations. 


104  MECHANICS-PROBLEMS. 

378.  A  uniform  ladder  weighing  loo  pounds  and 
52  feet  long  is  inclined  at  an  angle  of  45°  with  a 
rough  vertical  wall  and  a  rough  horizontal  plane.  If 
the  coefficient  of  friction  is  at  each  end  |,  how  far  up 
the  ladder  can  a  man  weighing  200  pounds  ascend 
before  the  ladder  begins  to  slip  ? 

379.  A  uniform  ladder  30  feet  long  is  equally  in- 
clined to  a  vertical  wall  and  the  horizontal  ground, 
both  rough  ;  a  man  with  a  hod  —  weight  224  pounds 
—  ascends  the  ladder  which  weighs  200  pounds. 
How  far  up  the  ladder  can  the  man  ascend  before  it 
slips,  the  tangent  of  the  angle  of  resistance  for  the 
wall  being  \  and  for  the  ground  \  ? 

380.  A  uniform  beam  rests  with  one  end  on  a 
rough  horizontal  plane,  and  the  other  against  a  rough 
vertical  wall,  and  when  inclined  to  the  horizon  at  an 
angle  of  30^  is  on  the  point  of  slipping  down  ;  sup- 
pose the  surfaces  equally  rough,  find  ya. 

381.  A  bolt  for  a  cylinder  head  has  8  threads  per 
inch;  mean  diameter  of  threads  i|-  inches,  average 
outside  diameter  of  nut  2|  inches,  inside  diameter  of 
bearing  surface,  1.6  inches.  The  nut  is  to  be  tight- 
ened by  a  pull  on  the  end  of  a  3-foot  wrench.  The 
coefificient  of  friction  for  threads  and  underneath  the 
nut  being  o.  1 5,  what  pull  should  be  exerted  in  order 
that  the  stress  in  the  bolt  shall  not  exceed  50  000 
pounds  ? 

Problems  pertaining  to  bolt  and  nut  friction  can  be  solved  by 
applying  the  combined  principles  of  Work  and  Friction.     Thus  for 


1/ 


FORCES  —  FRICTION 


105 


the  above  problem  suppose  that  the  specified  conditions  should  exist 
for  one  revolution.  This  involves  no  approximation,  simply  a  con- 
venience in  numerical  figures  which  otherwise  would  have  to  be 
divided  by  perhaps  a  hundred  or  thousand  to  apply  to  a  fractional 
part  of  a  revolution.     Then, 

Work       =  Work        4-  Work        +  Work 

on  wrench  on  threads  under  nut  of  lifting 

The  values  of  work  on  threads  and  work  under  nut  can  be  deter- 
mined near  enough  for  ordinary  cases  by  slight  approximations. 
As  shown  by  Fig.  59,  \V  =  .99  R,  or  usually  R  may  be  taken  equal 
to  W  also  the  length  of  thread  developed  (for  one  revolution)  ir  x 
\\" \  and  in  Ffg.  60  the  circumference  C  is  one  that  can  be  deter- 
mined by  the  condition  that  the  work  done  by  the  friction  of  all  the 
particles  outside  is  the  same  as  that  done  by  all  the  particles   inside. 

Its  radius  for  a  section  like  Fig.  60  is  x  =    -^-^ j-  ,  or,  for  this 

case,  X  =  I.I  I  inches,  which  is  approximately  the   same  as  would 


TT  X  1  J  in. 

As  the  thread  advances 
it  acts  like  a  wedge. 

Fig.  59. 


Fig.  60. 


result  by  taking  a  mean  circumference  between  1.6  inches  diameter 
and  2.75  —  or  1 .09  inches.  Therefore  ordinarily  use  the  mean  cir- 
cumference for  the  position  of  friction  under  nut. 

The  equation  for  Work  thus  becomes  : 

For  one  revolution. 

Wrench  Threads  Nut 

P(3X  i2X2X3l)  =  5ooooxo.i5X4.7i  +  5ooooxo.i5X  1.09x2x3]^ 

Lifting 

+  50  000  X  0.125. 

From  which  equation  find  P,  the  pull  that  should  be  exerted  on 
wrench  to  produce  50  000  pounds  stress  in  the  bolt. 


1 06  ME  CHANICS-PROBLEMS. 

382.  By  trial  in  a  60  000-pound  testing  machine 
we  have  obtained  with  a  builder's  lifting-jack  a  stress 
on  the  machine  of  6  000  pounds  for  a  certain  pull  on 
the  end  of  an  1 8-inch  bar.  What  was  that  pull  ? 
Mean  diameter  of  threads  was  1.50  inches,  there  were 
3  threads  to  the  inch,  and  diameter  of  bearing  that 
corresponds  to  the  mean  circumference  of  nut  de- 
scribed under  problem  381  was  1.78  inches.  Coeffi- 
cient of  friction  for  threads  was  0.15,  and  for  bearing 
0.15. 

383.  A  locomotive  bolt  has  10  threads  to  the  inch; 
mean  diameter  2  inches,  average  outside  diameter  of 
nut  4|-  inches,  diameter  of  hole  in  washer  on  which 
nut  turns  2.2  inches.  If  length  of  wrench  was  5 
feet,  pull  367  pounds,  and  stress  40  000  pounds,  what 
was  the  value  of  the  coefficient  of  friction  1 

384.  A  test  of  rope  friction  in  our  engineering  lab- 
oratory at  Tufts  College  has  given  the  following 
result : 

(The  weight  Tj  just  moving,  and  pull  T^  resisting  any  increased 
motion.     See  Fig.  6i.) 


For 

Weight  of 

T. 

=  100 

Pounds. 

Pull 

Ratio 

Pull 

Ratio 

Number  of 

T2 

Ti 

Number  of 

To 

Ti 

Laps 

Lbs. 

T2 

Laps 

Lbs. 

Ta 

\ 

81 

1.23 

If 

14 

\ 

65 

1-54 

II 

I 

I 

45 
32 

2i 

8 

5 

li 

25 

2\ 

A\ 

A 

19 

3 

Z\ 

Compute  the  ratios  of  T,  and  T^,  then  plot  the  results,  using  a 
scale  of  I  inch  =  i  lap  for  vertical  ordinates,  and  i  inch  =  ratio  of 


FOKCES—  FRICTION. 


07 


10  for  horizontal.  Sketch  the  most  probable  curve  for  the  plotted 
points,  observing  that  it  does  not  necessarily  pass  through  the  last 
point,  and  determine  whether  or  not  it  should  pass  through  the 
origin. 


Fig.  61. 


385.    Now  plot  the  same  results  on    the  specially 
ruled  paper  of  page  108. 


CThis  form  of  ruling  was  used  by  the  Burr-IIering-Freeman 
Commission  on  Additional  Water  Supply  of  the  city  of  New  York 
(1903)  for  plotting  wide  ranges  of  values  in  a  small  space  [7  million 
to  40  million  in  a  i^-inch  space],  yet  affMiding  increased  scale  for 
the  small  values.  It  has  not,  to  my  knowledge,  been  used  before 
for  mechanics'  problems  of  this  sort.) 


Hr^ 


Number  of  Laps 


FORCES.  —FRICTION. 


109 


Determine  how  this  form  of  ruHng  is  constructed. 
Plot  the  points  for  laps  and  ratios,  and  draw  the  most 
probable  straight  line  through  the  points  as  before. 
Should  the  line  pass  through  the  origin  .?  How  do 
the  points  for  i\  laps  and  2|  compare  on  this  straight 
line  with  those  on  the  curved  line  previously  plotted  } 

386.  The  lines  plotted  for  the  preceding  prob- 
lems are  sufficient  to  answer  directly  many  questions 
pertaining  to  that  particular  rope  and  piece  of  timber. 
For  the  same  conditions,  how  many  laps  are  needed 
to  hold  a  weight  of  300  pounds  with  a  pull  of  40 
pounds } 

387.  For  the  same  conditions,  with  2  laps  and  a 
pull  of  100  pounds,  what  weight  could  be  lowered 
into  the  hold  of  a  vessel  t 

388.  It  is  evident  that  the  plotted  lines  of  the  pre- 
ceding problems  would  not  apply  to  other  cases  of 
friction.  The  value  of  /a  the  coefficient  of  friction 
is  contained  in  the  equation  of  the  curves,  but  can- 
not yet  be  specified. 

For  the  purpose  of  finding  the  value  of  the  coefficient,  as  will  be 
done  later  on,  it  is  advisable  to  determine  the  slope  and  position  of 
the  plotted  line ;  that  is,  its  equation.     Notice  that  «  (the  number 

T, 

of  laps)  =  c  (a  constant  depending  on  the  slope  of  the  line)  x  „- 

(the  ratio  of  the  two  tensions).  The  value  of  c  (the  slope  of  the 
line)  would  depend  upon  the  stiffness  of  rope  and  roughness  of  the 
rubbing  surfaces.  For  this  particular  rope  and  piece  of  wood,  the 
value  of  c,  according  to  one  plotting  that  I  have,  is  2.08. 

What  does  your  plotting  indicate  for  the  above  ex- 
periment ? 


V 


lO 


ME  CHANICS  -^  PROBLEMS. 


389.    From  the  above  the  equation  of  the  line  be- 
comes   ;/  =z  2.08    los:^.     Write  your  equation  and 


T 


transpose  so  as  to  write  the  value  of  log  T,,  which  I 
find  to  be  log  T^  =  log  T^  +  .437  X  n.  Now  this 
equation  derived  from  the  laboratory  experiment  will 
be  seen  to  bear  a  close  relation  to  the  general  for- 
mula for  rope  and  belt  friction  which  will  now  be 
developed. 

The  author's  method  of  analysis  is  introduced  here  for  the  reason 
that  he  believes  it  to  be  more  easily  understood  than  the  methods 
usually  presented  in  text  books. 

At  B  there  would  be  a  tension  of  Tj,  at  A,  T,„  and  at  any  point, 
C,  there  would  be  two  tensions,  T  and  T  +  rt'T.  Now  if  we  knew 
the  reaction  at  the  point  C  we  could  multiply  it  by  /*  and  obtain  the 


friction.  To  find  this  reaction  R  draw  the  parallelogram  of  force, 
Fig.  63.  Let  an  arc  of  the  angle  dd  measured  at  unit  distance  from 
the  center  be  arc  dQ,  then  at  distance  T  the  arc  w^ould  be  T  xarc  dQ, 
and  for  a  very  small  angle  —  a  differential  angle  —  this  value  of  the 
arc  would  equal  R. 

R  =  T  arc  dd 
/iR  =  |uT  arc  dd. 


FOR  CES  —  FRIC  TION  1 1  I 

Now  the  friction  «R  =  the  difference  in  tensions  t/T. 
dT  =  AtT  arc  dB. 

That  is,  for  an  infinitesimal  arc,  the  difference  in  tension  =  uY  x 
the  infinitesimal  arc.  If  we  take  a  very  large  number  of  small  arcs 
we  can  find  the  friction  at  each  point,  add  up  and  get  the  sum  total 
of  friction  ;  or,  summing  up  by  the  calculus. 


r  ¥-'/'-" 


T 

log,l\    -    l0g.T2    =/X^ 

Now  6  =  2  rrn  in  which  n  is  the  number  of  laps  ;  and  the  Naperian 
log  can  be  changed  to  the  common  system  by  multiplying  by  .4343. 

•••  logio  T,  -  log,o  T2  =  .4343  ^X  2irn 
logTj=logT2+  2.7288  M^/ 

which  is  the  general  formula  for  rope  and  belt  friction.     It  contains 
variable  quantities  :  the  tension  Tj,  tension  T^,  and  the  number  of 
laps  //.     Any  two  of  these  being  given  the  third  can  be  found. 
The  formula  deduced  by  experiment  in  problem  389  was^ 

log  T,  =  log  T,  4-  .437  «• 
The  similarity  of  this  with  the  general  formula  is  evident.     The 
last  term  must   contain  the  value  of  fi  the  coefficient  of  friction. 
To  find  this  value  solve  the  two  equations,  and  we  have 
2.7288 /x«  =  .437  n 
/x  =  o.  1 5 

Then  if  this  be  taken  as  the  vakie  of  11,  how  many 
laps  would  be  necessary,  according  to  the  general 
formula,  for  100  pounds  to  just  move  14  pounds.^ 
(Check  result  with  data  of  problem  384.) 

390.  A  weight  of  100  pounds  just  m.oves  37 
pounds,  both  being  connected  by  a  plain  leather  belt 
that  encircles  one-half  of  a  14-inch  iron  pulley  that 
does  not  turn.  Plot  the  point  on  the  paper  of  page 
108  ;  draw  the  Hne  of  friction,  and  write  the  equation 
of  the  line.     Then  compare  with  the  general  formula 


112 


ME  CHANICS-PR  OBLEMS. 


and  determine  the  value  of  the  coefficient  of  friction 
for  the  plain  belt  and  iron  pulley. 


...  .iJi 

Fig.  64. 

391.  In  the  same  way  as  by  problem  390,  after  the 
belt  had  been  treated  -with  "  cling-fast  "  belt  dressing 
55  pounds  just  moved  13.  Plot  the  line  and  find  the 
coefficient  of  friction. 

Further  application  of  the  general  belt  and  rope  friction  formula 
is  seen  in  the  problems  that  follow. 

392.  According  to  conditions  of  friction  as  in  prob- 
lem 390,  how  many  turns  would  have  to  be  taken 
around  a  capstan  in  order  to  lower  a  barrel  of  salt, 
150  pounds,  into  a  dory  without  pulling  over  50 
pounds  ? 


FORCES— FRICTION,  II3 

393.  A  weight  of  5  tons  is  to  be  raised  from  the 
hold  of  a  steamer  by  means  of  a  rope  which  takes 
3^^  turns  around  the  drum  of  a  steam-windlass.  If 
/A  =  0.234,  what  force  must  a  man  exert  on  the  other 
end  of  the  rope  t 

394.  A  man  by  taking  2\  turns  around  a  post  with 
a  rope,  and  holding  back  with  a  force  of  200  pounds, 
just  keeps  the  rope  from  surging.  Supposing  /x  = 
0.168,  find  the  tension  at  the  other  end  of  the  rope. 

395.  A  leather  belt  will  stand  a  pull  of  200  pounds. 
It  passes  around  one-half  the  circumference  of  a 
pulley  that 'is  4  feet  in  diameter  and  making  150 
revolutions  per  minute.  What  power  will  it  transmit 
if  the  coefificient  of  friction  between  the  belt  and 
pulley  is  o.  i  '^. 

396.  A  belt  laps  150°  around  a  3-foot  pulley,  mak- 
ing   130  revolutions    per  minute;  the  coefficient  of    ^ 
friction  is  0.35.     What  is  the  maximum  pull  on  the 
belt  when  20  horse-power  is  being  transmitted  and 
the  belt  is  just  on  the  point  of  slipping.? 

397.  A  weight  of  2  000  pounds  is  to  be  lowered 
into  the  hold  of  a  ship  by  means  of  a  rope  which 
passes  over  and  around  a  spar  lashed  across  the  hatch- 
coamings  so  as  to  have  an  arc  of  contact  of  i  \  cir- 
cumferences. If  /i  =  2^,  what  force  must  a  man 
exert  at  the  end  of  the  rope  to  control  the  weight } 

398.  A  hawser  is  subjected  to  a  stress  of   10  000 
pounds.     How  many  turns  must  be  taken  around  the    // 
bitts,  in  order  that  a  man  who  cannot  pull  more  than 


114  MECHANICS-PROBLEMS. 

250  pounds    may  keep   it   from    surging,  supposing 
ft  =  0.168  ? 

399.  A  rope  drive  carrying  20  ropes  has  a  pulley 
16  feet  in  diameter,  and  transmits  600  horse-power 
when  running  at  90  revolutions  per  minute.  Taking 
At  =  0.7  and  the  angle  of  contact  180°,  find  the  ten- 
sions on  the  tight  and  slack  sides  of  the  ropes. 

From  the  data  that  is  given  find  by  the  principles  of  Work  the 
force  that  the  belt  is  transmitting.     It  is 

Ti-T2=  218.8  pounds 
Substitute  this  value  of  Tg  in  the  general  formula,  also  the  value 
of  \i.  and  71 ;  then 

log  T,  ==  log  (Tj- 2 18:8) +  .9550     * 

^"^(t, -^218.8)  =  -955 

T. 

9.02 


1\- 218.8 
From  which  find  T^ ;  then  find  T2. 

400.  A  belt  for  a  dynamo  is  to  encircle  half  of  an 
18-inch  pulley.  The  speed  of  pulley  is  to  be  1060 
revolutions  per  minute  ;  horse-power  to  be  trans- 
mitted, 100  ;  coefficient  of  friction,  0.2  :  thickness  of 
belt  to  be  ||  inches,  and  working  strength  300 
pounds  per  square  inch.     What  should  be  its  width  } 

401.  A  main  driving  belt  is  to  encircle  half  of  a 
54-inch  pulley.  The  speed  of  pulley  is  to  be  350 
revolutions  per  minute  ;  horse-power  transmitted,  520  ; 
coefficient  of  friction,  0.2.  Thickness  of  belt,  accord- 
ing to  specifications,  is  to  be  ||  inches,  and  work- 
ing strength  450  pounds  per  square  inch.  What 
should  be  its  width  ? 


FOR  CES  —  FRIC  TION.  I  I  5 

402.  A  plain  belt  without  dressing  encircling  one- 
half  of  a  pulley,  when  just  on  the  jDoint  of  slipping 
has  a  tension  of  i  ooo  pounds  on  the  taut  side.  More 
machinery  being  put  into  use,  rosin  is  thrown  on  the 
belt  If  the  tension  on  the.  slack  side  remains  the 
same  as  "before,  and  the  belt  is  just  on  the  point  of 
slipping,  what  horse-power  will  be  transmitted,  diame- 
ter of  pulley  being  lO  feet,  and  revolutions  per  min- 
ute, 140  .^ 

403.  A  single  fixed  pulley,  6  inches  in  diameter, 
turns  on  an  axle  2  inches  in  diameter  ;  coefficient  of 
friction,  0.2.  A  weight  of  500  pounds 
is  lifted  by  means  of  this  pulley.  Find 
the  force  V  that  is  required. 


Friction  causes  the  axle  to  creep,  as  it  were,  on  its 
bearings.     S  moves  a  little  off  center,  coming  nearer 
to  P.     When  the  value  of  K  can  be  found  the  fric-.         ^i?-  65. 
tion  will  be  determined  by  multiplying  by  yu.     To  find  R  : 

S   =    R2   +  /i2K2 

as  will  be  evident  by  plotting  a  parallelogram  of  force. 


S  = 

P  +  W 

•.  R  = 

P+  W 

Vi  +  /*' 

= 

p-h  500 
1.02 

/iR  = 

P+=^oo 

0.2  X ^— 

1.02 

Now  to 

findP: 

Take  moments  about  C,  the  center 

-  P  X  3  +  500  X  3  +  /xR  X  I 
P  =  570  pounds. 


Il6  MECITANICS-PROBLEMS. 

404.  A  shaft  makes  50  revolutions  per  minute. 
The  load  on  the  bearing  is  8  tons,  the  diameter  of  the 
bearing  is  7  inches,  and  the  average  coefficient  of 
friction  is  0.05.  At  what  rate  is  heat  being  gen- 
erated ? 

S  =  P  -f  W 
=  8  tons 
P  +  W 


R 


=  15980 

ftR  =  799  pounds 
Work      =  force  X  distance 

of  friction 

=  799  X  (1^0  X  2  X  ---^  X  50) 

=  73  500  foot-pounds  per  minute. 

405.  A  single  fixed  pulley,  2  feet  in  radius,  turns 
on  an  axle  i  inch  in  radius ;  the  weight  of  the  pulley 
is  80  pounds.  A  weight  of  500  pounds  is  lifted  by 
means  of  this  pulley.  What  force  P  is  required  } 
The  coefficient  of  friction  between  axle  and  bearing 
is  o.  I  ;  the  rope  is  flexible,  and  without  weight,  and 
P  acts  vertically. 

406.  Find  the  horse-power  necessary  to  turn  a  shaft 
9  inches  in  diameter  making  75  revolutions  per  min- 
ute, if  the  total  load  on  it  is  12  tons  and  11  =  .015. 

407.  Let  P  and  W  be  inclined  to  each  other  at  an 
angle  of  90°  ;  radius  of  pulley  is  6  inches ;  radius  of 
axle  I  inch  ;  coefficient  of  friction,  0.2.  Determine 
the  relation  of  P  and  W  in  case  of  incipient  motion. 


V 


FORCES  —  FRICTION.  I  1 7 

408.  A  horizontal  axle  10  inches  in  diameter  has  a 
vertical  load  upon  it  of  20  tons,  and  a  horizontal  pull 
of  4  tons.  The  coefficient  of  friction  is  0.02.  Find 
the  heat  generated  per  minute,  and  the  horse-power  ^ 
wasted  in  friction,  when  making  50  revolutions  per 
minute. 

409.  The  shaft  of  a  i  000-kilowatt  dynamo  is  25 
inches  in  diameter,  makes  ibo  revolutions  per  min- 
ute, and  carries  a  total  load  of  45  000  pounds.  The 
coefficient  of  friction  being  0.05,  find  the  horse-power 
lost  in  heat  that  is  generated  by  friction. 

410.  Find  the  horse-power  absorbed  in  overcoming 
the  friction  of  a  foot-step  bearing  with  flat  end  4  inches     / 
in  diameter,  the  total  load  being  i^  tons,  the  number 
of  revolutions  100  per  minute,  and  the  average  coeffi- 
cient of  friction  0.07. 

force      X     distance 

Work      =  \\>  X  (fj  X  {-.^)  X  2 TT  X  100. 

of  friction 

The  distance  being  obtained  by  consideinng  a  circumference  as  in 
problem  381,  outside  of  which  the  work  is  the  same  as  that  inside. 
For  a  bearing  with  a  flat  end  that  circumference  has  a  radius  of  two- 
thirds  of  R. 

411.  Calculate  the  horse-power  absorbed  by  a  foot- 
step bearing  with  flat  end  8  inches  in  diameter  when 
supporting  a  load  of  4  000  pounds,  and  making  100 
revolutions  per  minute,  coefficient  of  friction  0.03. 

412.  A   1 50-horse-power  turbine  has  an  oak  step 

6  inches  in  diameter  and  with  conical  end  tapering  ^ 
45°.     If   the    load  on   the  step  be    2  tons,  and  the 


1 1  8  ME  CIIA  NICS-PR  OBLEMS. 

coefficient  of  friction  between  the  wood  and  its  metal 
seat  be  0.3,  find  the  horse-power  thus  absorbed  at 
65  revolutions  per  minute. 

To  resist  the  load  of  2  ^tons  would  require  a  pressure  of  2.83  tons 
by  the  45°  slope  of  the  foot-step.  The  mean  circumference  would 
be  as  in  preceding  problems,  distant  two-thirds  Rfrom  center. 

413.  The  shaft  of  a  vertical  steam  turbine  has  a 
conical  foot-step  bearing  3.5  inches  in  diameter,  and 
length  3  inches.  Total  load  on  shaft,  i  500  pounds  ; 
speed  2  500  revolutions  per  minute ;  coefficient  of 
friction,  0.07.  Find  the  horse-power  that  tends  to 
"  burn  out  "  the  foot-step. 


MO  TION. 


119 


III.    MOTION 

414.  A  body  moving  with  a  velocity  of  5  feet  per 
second  is  acted  on  by  a  force  which  produces  a  con- 
stant acceleration  of  3  feet  per  second.  What  is  the 
velocity  at  the  end  of  20  seconds } 

Velocity  gained  =  acceleration  per  second  x  number  of 
seconds. 

V  —  a  y.  t 
=  3  X  20 

=  60  feet  per  second. 
Final  velocity  =  60-1-5 

=  65  feet  per  second. 

415.  The  initial  velocity  of  a  stone  is  12  feet  per 
second ;  this  velocity  decreases  uniformly  at  the  rate 
of  2  feet  per  second.  How  far  will  the  stone  have 
traveled  in  5  seconds } 

416.  Two  trains  A  and  B  moving  towards  each 
other  on  parallel  rails  at  the  rate  of  30  miles  and 
45  miles  an  hour,  are  5  miles  apart  at  a  given  instant. 
How  far  apart  will  they  be  at  the  end  of  6  minutes 
from  that  instant,  and  at  what  distances  are  they  from 
the  first  position  of  A  t 

417.  Two  trains,  130  and  no  feet  long,  pass  each 
other  in  4  seconds  when  going  in  opposite  directions. 
The  velocity  of  the  longest  train  being  double  that  of  1/ 
the  other,  find  at  what  speed  per  hour  each  is  going. 


Motion, 


121 


418.  Two  trains  going  in  opposite  directions  pass 
each  other  in  3  seconds.  One  train  is  142  feet  long 
and  the  other  88  feet  long.  When  going  in  the  same 
direction  one  passes  the  other  in  1 5  seconds.  How 
fast  is  each  train  going  ? 

419.  The  velocity  of  a  train  is  known  to  have  been 
increasing  uniformly;  at  one  o'clock  it  was  12  miles 
per  hour ;  at  10  minutes  past  one  it  was  36  miles  per 
hour.     What  was  it  at  yl  minutes  past  one.** 

420.  A  train  moving  at  the  rate  of  30  miles  an 
hour  is  brought  to  rest  in  2  minutes.  The  retarda- 
tion is  uniform.     How  far  did  it  travel } 

A  railroad  train  is  moving  at  30  miles  an  hour.  In  each  second, 
then,  it  moves  44  feet.  Its  velocity  for  each  second  during  the  time 
/  may  be  represented  as  in  Fig.  66  by  lines  of  equal  length,  and  the 
area  of  the  rectangle,  or  vt,  represents  the  distance  passed  over. 
This  is  an  illustration  of  uniform  motion. 

When  a  railroad  train  starts  from  a  station,  and  by  uniform  gain 
in  speed  attains  a  velocity  of  30  miles  an  hour,  the  distance  passed 


illl 

]Arta=vt 


Fig.  66.  Fig.  67.  Fig.  68. 

over  may  be  graphically  represented  as  in  Fig.  67.     The  area  would 
represent  said  distance. 

421.  Similarly,  what  condition  of  speed  of  the  rail- 
road train  would  Fig.  6'^  represent } 

Note  that  the  area  of  Fig.  68  is  v^f  -^  \  {v  —  v^  t,  or  vj  +  \  afi. 

422.  A    stone  skimming   on  ice  passes   a  certain 
point  with  a  velocity  of  20  feet  per  second,  then  suf- 


1 2  2  ME  CHA  NICS-PR  OBL  EMS. 

fers  a  retardation  of  one  unit.  Find  the  space  passed 
over  in  the  next  lo  seconds,  and  the  whole  space 
traversed  when  the  stone  had  come  to  rest. 

423.  On  the  New  York  Central  and  Hudson  River 
Railroad  test  tracks  near  Schenectady,  an  electric  loco- 
motive hauled  9  Pullman  cars  at  a  running  speed  of 
60  miles  per  hour.  The  average  acceleration  from 
start  to  full  speed  was  0.5  miles  per  hour  per  second. 
The  retardation  on  applying  air  brakes  was  0.88  feet 
per  second  per  second.  These  results  were  obtained 
by  carefully  timing  the  train  at  measured  stations. 
Total  distance  was  4  miles.    What  was  the  total  time  .'* 

"  ^  424.  A  train  is  running  at  the  rate  of  60  miles  an 
hour  when  the  steam  is  turned  off  ;  it  then  runs  on  a 
level  track  for  3 J  miles  before  stopping.  If  friction 
be  the  constant  retarding  force,  find  its  value  in 
pounds  per  ton.  Also  how  far  does  the  train  run  in 
3  minutes  from  the  instant  steam  is  turned  off } 

425.  A  body  acted  on  by  a  constant  force  begins 
to  move  from  a  state  of  rest.  It  is  observed  to  move 
through  55  feet  in  a  certain  2  seconds,  and  through 
yy  feet  in  the  next  2  seconds.  What  distance  did  it 
describe  in  the  first  6  seconds  of  its  motion  1 

426.  A  steamer  approaching  a  wharf  with  engines 
reversed  so  as  to  produce  a  uniform  retardation  is 
observed  to  make  500  feet  during  the  first  30  seconds 
of  the  retarded  motion  and  200  feet  during  the  next 
30  seconds.  In  how  many  more  seconds  will  the 
headway  be  completely  stopped  ? 


MOTION.  123 

427.  Two  bodies  are  let  fall  from  the  same  point 
at  an  interval  of  2  seconds.  Find  the  distance  be- 
tween them  after  the  first  has  fallen  for  6  seconds. 

For  I  St  body,  s  =\gf' 

=  ^  X  32  X  6- 

=  576  feet 
For  2d  body,  s  =  \gf^ 

=  Kx  32  X  4' 
=  256  feet 
.'.  distance  apart  =  576-256 
=  320  feet. 

428.  A  stone  is  projected  vertically  upwards  with 

a  velocity  of  80  feet  per  second  from  the  summit  of       A- 
a  tower  96  feet  high.     In  what  time  will  it  reach  the 
ground,  and  with  what  velocity  ? 

429.  A  hammer  of  10  tons  weight  falling  from  a 
height  of  4  feet  drives  a  wooden  pile  and  comes  to 
rest  in  ^J^-  second.  How  far  does  it  drive  the  pile  ? 
And  assuming  the  force  is  uniform  find  it. 

430.  A  stone  is  dropped  into  a  well,  and  the  sound 
of  its  striking  is  heard  2y'2  seconds  after  it  is  dropped  ; 
the  velocity  of  sound  in  air  is  i  200  feet  per  second. 
What  is  the  depth  of  the  well  .-^ 

Let  s  =  depth  of  well. 

.'.  time  for  sound  to  come  up  = seconds. 

I  200 

Tinie  for  stone  to  fall  is  found  from  formula 
s^  }ygr- 

^        2S         s  J  ,         ^ 

.*.  ^  =  —  =  — r  J        and        /  =  —  . 
^        i^  4 


1 24  MECHANICS-PROnLEMS. 


Time  for  stone  to  fall  +  time  for  sound  to  come  up  =  2y' 


I  200         4         12 
s  +  300  ys  =  ^  100 
s  ±  150-  4-  300  Vi-  =  3  100  ±  150^ 
Vi-  =  —  310  an  inadmissible  value, 
or  \/s  =  +  10 

.*.     s  =  100  feet,  depth  of  well. 

431.  A  stone  is  dropped  from  a  tower  of  height  a 
feet ;  another  is  projected  upwards  vertically  from  the 
foot  of  the  tower ;  the  two  start  at  the  same  moment. 
What  is  the  initial  velocity  of  the  second  if  they  meet 
halfway  up  the  tower  .? 

432.  A  stone  is  dropped  into  a  well,  and  the  sound 
of  the  splash  is  heard  y.y  seconds  afterwards.  Find 
the  distance  to  surface  of  the  water,  supposing  the 
velocity  of  sound  to  be  i  120  feet  per  second. 

433.  A  bucket  is  dropped  into  a  well  and  in  4  sec- 
onds the  sound  of  its  striking  the  water  is  heard. 
How  far  did  the  bucket  drop  ? 

434.  A  balloon  has  been  ascending  vertically  at  a 
uniform  rate  of  4^  seconds,  and  a  test  ball  dropped 
from  it  reaches  the  ground  in  7  seconds.  Find  the 
velocity  of  the  balloon  and  the  height  from  which 
the  ball  was  dropped. 

435.  From  a  balloon  that  is  ascending  with  velocity 
of  32  feet  per  second,  a  ball  drops  and  reaches  the 
ground  in  17  seconds.     How  far  up  is  the  balloon  ? 

4389 


MOTION. 


125 


436.  A  ball  is  let  fall  to  the  ground  from  a  certain 
height,  and  at  the  same  time  another  ball  is  thrown 
upwards  with  just  sufficient  velocity  to  carry  it  to  the      1/ 
point  from  which  the  first  one  fell.    When  and  where 
will  they  meet? 

437.  A  cake  of  ice  slides  down  a  smooth  chute 
that  is  set  at  an  angle  of  30°  to  the  horizon.  Through 
how  many  feet  vertically  will  the  cake  of  ice  fall  in 
the  fourth  second  of  its  motion? 

The  acceleration  for  a  body  falling  vertically  is^,  32  feet  per  sec- 
ond per  second.  The  acceleration  component  measured  along  a 
30°-plane  is  32  x  sin  30°,  or  16  feet  per  second  per  second. 

=  72  feet,  for  3  seconds 
=  128  feet  for  4  seconds 
Therefore  space  along  plane  in  the  4th  second 
=  56  feet 

438.  A  cable  car  "•  runs  wild  "  down  a  smooth  track 

of  inclination  20   to  the  horizontal.     How  far  does  it     ^' 
go  during  the  first  8  seconds  after  starting  from  rest  ? 

439.  A  body  is  projected  up  a  plan  of  30°  incli- 
nation with  a  velocity  of  80  feet  per  second.  How 
long  before  it  will  come  to  rest  ?  How  far  will  it 
go  up  the  plane. 

440.  A  body  is  sliding  with  velocity  u  down  an  in- 
clined plane  whose  inclination  to  the  horizon  is  30°.    - 
Find  the  horizontal  and  vertical  components  of  this 
velocity. 

441.  A  stone  was  thrown  with  a  velocity  of  33  feet 
per  second  at  right  angles  to  a  train  that  was  going 


1 26  MECHANICS-PROBLEMS. 

30  miles  an  hour.  It  hit  a  passenger  who  was  sitting 
on  the  opposite  side  of  the  car  that  was  9  feet  wide. 
How  far  in  front  of  him  should  be  the  hole  in  the 
window  .-* 

442.  A  deer  running  at  the  rate  of  20  miles  an 
hour  keeps  200  yards  distant  from  a  sportsman.  How 
many  feet  in  front  of  the  deer  should  aim  be  taken  if 
the  velocity  of  the  bullet  be  i  000  feet  per  second  t 

443.  A  boat  is  rowed  at  the  rate 
of  5  miles  an  hour  on  a  river  that 
runs  4  miles  an  hour.  In  what  di- 
V  rection  must  the  boat  be  pointed 
to  cross  the  river  perpendicularly  .-* 
With  what  velocity  does  it  move  t 

Let    OX    be  4  units  in    length  to    represent 
the  velocity  of  the  stream. 
Draw  OM  perpendicular  to  OX.     The  resultant  velocity  is  to  be 
in  the  direction  OM. 

With  center  X  and  radius  of  5  units  describe  an  arc  cutting  OM 
in  P. 

Join  XP,  and  complete  the  parallelogram  of  velocities  OXPQ. 

OQ  is  the  required  direction. 
The  angle  QOP  =  sin-'  i. 
Therefore  the  boat  must  not  be  rowed  straight  across,  but  up 
stream  at  an  angle  of  53°  10'. 
To  find  the  resultant  velocity  : 

OP2  =  OQ'  -  QP2 

=  5^-4^ 
=  25-16 

=  9 

•■•.  ^^  =  ^ 
.*.  the  boat  crosses  the  river  at  the  rate  of  3  miles  an  hour. 


MOTION,  127 

444.  A  river  flows  at  the  rate  of  2  miles  per  hour. 
A  boat  is  rowed  in  such  a  way  that  in  still  water  its 
velocity  would  be  5  feet  per  second  in  a  straight  line. 
The  river  is  3  000  feet  wide  ;  the  boat  starting  from 
one  shore,  is  headed  60°  up-stream.  Where  will  it 
strike  the  opposite  shore  } 

445.  A  bullet  moving  upwards  with  a  velocity  of 
I  000  feet  per  second,  hits  a  balloon  rising  with 
velocity  100  feet  per  second.  Find  the  relative 
velocity. 

446.  A  train  at  45  miles  an  hour,  passes  a  carriage 
moving  10  yards  a  second  in  the  same  direction  along 
a  parallel  road.     Find  the  relative  velocity. 

447.  To  a  passenger  in  a  train,  raindrops  seem  to 
be  falling  at  an  angle  of  30°  to  the  vertical ;  they  are 
really  falling  vertically,  with  velocity  80  feet  per 
second.     What  is  the  speed  of  the  train  t 

448.  Two  roads  cross  at  right  angles ;  along  one 
a  man  walks  northward  at  4  miles  per  hour,  along  the 
other  a  carriage  goes  at  8  miles  per  hour.  What  is 
the  velocity  of  the  man  relative  to  the  carriage  } 

449.  A  steamer  is  going  east  with  a  velocity  of 
6  miles  per  hour ;  the  wind  appears  to  blow  from  the 
north  ;  the  steamer  increases  its  velocity  to  1 2  miles 
per  hour,  and  the  wind  now  appears  to  blow  from  the 
north-east.  What  is  the  true  direction  of  the  wind 
and  its  velocity } 


1 28  MECHANICS-PROBLEMS. 

450.  A  ship  is  sailing  north-east  with  a  velocity  of 
10  miles  per  hour,  and  to  a  passenger  on  board  the 
wind  appears  to  blow  from  the  north  with  a  velocity 
of  10  V2  miles  per  hour.  Find  the  true  velocity  of 
the  wind. 

451.  A  fly-wheel  revolves  1 2  times  a  second.  What 
is  the  angular  velocity  of  a  point  on  the  rim  taken 
about  the  center  } 

452.  A  broken  casting  flies  along  a  concrete  floor 
with  initial  velocity  of  50  feet  per  second.  The 
coefficient  of  friction  being  \  what  will  be  its  velocity 
after  3  seconds } 

One  of  the  axioms  for  problems  in  Motion  is,  that 
P  the  force  :  W  the  weight  =^  a  :  g. 
The  force  producmg  motion  :  the  total  weight  moved  =  the  accel- 
eration produced  by  the  force  :  the  acceleration  that  gravity  would 
produce. 

For  the  above  example  the  force  producing  motion  (or  in  this 
case  retardation)  is  W  x  §  and 

W  X  J  ;  W  =  «  :  32 

a  =  16  feet  per  second  per  second 
After  3  seconds  the  velocity  would  be 

z'  =  50  —  16  X  3 
=  2  feet  per  second. 

453.  A  locomotive  that  weighs  100  tons  is  increas- 
ing its  speed  at  the  rate  of  100  feet  a  minute.  What 
is  the  effective  force  acting  on  it .? 

454.  An  ice  boat  that  weighs  i  000  pounds  is 
driven  for  30  seconds  from  rest  by  a  wind  force  of 
100  pounds.  Find  the  velocity  acquired  and  the 
distance  passed  over. 


/ 


MO  TJON.  I  29 

455.  A  5 -pound  curling  iron  is  thrown  along  rough 
ice  against  a  friction  of  one-fifth  of  its  weight ;  it 
comes  to  rest  after  going  a  distance  of  40  feet.  What 
must  have  been  its  velocity  at  the  beginning  ? 

456.  The  table  of  a  box-machine  weighs  50  pounds 
and  is  pulled  back  to  its  starting  position,  a  distance  of 
6  feet,  by  a  falling  weight  of  20  pounds.  What  time, 
neglecting  friction,  will  thus  be  used  in  return 
motion } 

457.  A  body  whose  mass  is  108  pounds  is  placed 
on  a  smooth  horizontal  plane,  and  under  the  action 

of  a  certain  force  describes  from  rest  a  distance  of    \/ 
1 1^  feet  in  5  seconds.     What  is  the  force  acting.? 

458.  Two  bodies  A  and  B,  that  weigh  50  pounds 
and  10  pounds,  are  connected  by  a  string  ;  B  is  placed 
on  a  smooth  table,  and  A  hangs  over  the  edge. 
When  A  has  fallen  10  feet,  what  is  the  accumulated 
work  of  the  bodies  jointly,  and  what  of  them  severally } 

459.  A  500-volt  electric  motor  imparts  velocity  to 
an  8-ton  car  so  that  at  the  end  of  20  seconds  it  is 
moving  on  a  level  track  at  the  rate  of  10  miles  an    ^ 
hour ;  the  total  efficiency  of  the  motor  and  car  is  60 
per  cent.     What  amperes  are  necessary  ? 

460.  Show  that  to  give  a  velocity  of  20  miles  an 
hour  to  a  train  requires  the  same  energy  as  to  lift  it 
vertically  through  a  height  of  13.4  feet. 

461.  What  force  must  be  exerted  by  an  engine  to 
move  a  train  of  weight  100  tons  with  10  units  of  accel- 
eration, if  frictional  resistances  are  5  pounds  per  ton  "i 


1 30  MECHANICS-PROBLEMS. 

'^  462.  A  train  that  weighs  60  tons  has  a  velocity  of 
40  miles  an  hour  at  the  time  its  power  is  shut  off. 
If  the  resistance  to  motion  is  10  pounds  per  ton,  and 
no  brakes  are  applied,  how  far  will  it  have  traveled 
when  the  velocity  has  reduced  to  10  miles  per  hour? 

The  retardation  a  will  be  found  to  be  o,i6  feet  per  second  per 
second  ;  the  total  loss  in  velocity  is  44  feet  per  second.  Then  find 
the  time,  and  lastly  the  space  by  observing  that  space  =  average 
velocity  X  time. 

463.  A  locomotive  running  on  a  level  track  brings 
a  train  of  weight  1 20  tons  to  a  speed  of  30  miles  an 
hour  in  2  minutes.  The  resistance  to  motion  of  the 
train  being  uniform  and  equal  to  8  pounds  per  ton, 
what  will  be  the  required  horse-power  at  the  draw-bar 
and  what  the  distance  from  the  starting  point  when 
the  speed  of  30  miles  an  hour  is  attained .? 

464.  A  freight  train  of  100  tons  weight  is  going 
at  the  rate  of  30  miles  an  hour  when  the  steam  is 
shut  off  and  the  brakes  applied  to  the  locomotive. 
Supposing  the  only  friction  is  that  at  the  locomotive, 
the  weight  of  which  is  20  tons,  what  is  the  coefficient 
of  friction  if  the  train  stops  after  going  2  miles .? 

20  X  /^  :  100  =  ^  (which  can  be  found  from  the  data 
given  in  the  problem)  -.32. 

^  465.  A  train  of  100  tons,  excluding  the  engine, 
runs  up  a  i  %  grade  with  an  acceleration  of  i  foot  per 
second.  If  the  friction  is  10  pounds  per  ton,  find 
the  pull  on  the  drawbar  between  engine  and  train. 

Total  force  =  force  for  acceleration  -{-  force  for  lifting 
+  force  for  friction. 


MOTION.  131 

466.  A  body  is  projected  with 
a  velocity  of  20  feet  per  second 
down  a  plane  whose  inclination  is 
25°  ;  the  coefficient  of  friction  be-  Fig.  70. 

ing  0.4.    Determine  the  space  traversed  in  2  seconds. 
P  :  W  =  ^  :  ^ 
(.423  -  -3625)  X  W  :  W  =  ^  :  ^. 
The  space  traversed, 

^  =  V/  -f-  \  af. 

467.  A  body  slides  down  a  rough  inclined  plane  100 
feet  long,  the  sine  of  whose  angle  of  inclination  is  0.6  ; 
the  coefficient  of  friction  is  \.  Find  the  velocity  at 
the  bottom.  If  projected  up  the  plane  with  a  velocity 
that  just  carries  it  to  the  top,  find  that  velocity.  / 
The  forces  acting  down  the  plane 

=  \V  X  sin  a  —  W  X  cos  a  X  ^. 

468.  An  electric  car  at  the  top  of  a  hill  becomes 
uncontrollable  and  ''  runs  wild  "  down  a  grade  of  i 
vertical  to  20  horizontal  a  distance  of  \  mile.  The 
resistance  to  friction  being  20  pounds  per  ton  and  the 
total  weight  of  car  and  passengers  50  tons,  how  fast 
will  the  car  be  going  when  it  reaches  the  foot  of  the 
hill  ? 

469.  Two  weights  of  120  and  100  pounds  are  sus- 
pended by  a  fine  thread  passing  over  a  fixed  pulley 
without  friction.  What  space  will  either  of  them  pass 
over  in  the  third  second  of  their  motion  from  rest } 

Observe  that  the  force  producing  motion  is  in  this  case  20  pounds, 
and  the  total  weight  moved  is  220  pounds.  Then  a  =2.92  feet  per 
second  per  second. 


I  3 2  MECHANICS-PROBLEMS. 

470.  A  man  who  is  just  strong  enough  to  Hft  150 
pounds  can  hft  a  barrel  of  flour  of  200  pounds  weight    y 
when  going  down  on  an  elevator.      How  fast  is  the 
velocity  of  elevator  increasing  per  second  ? 

471.  A  cord  passing  over  a  smooth  pulley  carries 
10  pounds  at  one  end  and  54  pounds  at  the  other. 
What  will  be  the  velocity  of  the  weight  5  seconds 
from  rest,  and  what  will  be  the  tension  in  the  cord } 

After  computing  the   acceleration  that  the   two  weights   would 
have,  find  the  equivalent  force,  or  tension,  that  would  be  required  to 
cause  said  acceleration  on  the  lo-pound  weight,  which  is  the  one 
that  is  being  moved.      We  have  a  ==  27,  and 
P  :  10  =  27  :  32 
P,  the  tension  =  8.4  pounds. 

472.  Two  strings  pass  over  a  smooth  pulley ;  on 
one  side  both  strings  are  attached  to  a  weight  of  5 
pounds,  on  the  other  side  one  string  is  attached  to  a 
weight  of  3  pounds,  the  other  to  one  of  4  pounds. 
Find  the  tensions  during  motion. 

473.  Weights  of  5  pounds  and  1 1  are  connected 
by  a  thread  ;  the  1 1 -pound  weight  is  placed  on  a  smooth 
horizontal  table,  while  the  other  hangs  over  the  edge. 
If  both  are  then  allowed  to  mpve  under  the  action  of 
gravity,  what  is  the  tension  of  the  thread } 

474.  A  lo-pound  weight  hangs  over  the  edge  of  a 
table  and  pulls  a  45-pound  box  along  ;  the  coefficient 
of  friction  between  the  table  and  the  box  is  0.5. 
Find  the  acceleration  and  the  tension  in  the  string. 

\)     475.    An  engine  draws  a  three-ton  cage  up  a  coal- 
pit shaft  at  a  speed  uniformly  increasing  at  the  rate 


MOTION,  133 

of  5  feet  per  second  in  each  second.     What   is  the 
tension  in  the  rope  ? 

476.  A  balloon  is  moving  upward  with  a  speed 
which  is  increasing  at  the  rate  of  4  feet  per  second  per 
second.  Find  how  much  the  weight  of  a  body  of  10 
pounds  as  tested  by  a  spring  balance  on  it,  would 
differ  from  its  weight  under  ordinary  circumstances. 

477.  An  elevator  of  300  pounds  weight  is  being 
lowered  down  a  coal  shaft  with  a  downward  accelera- 
tion of  5  feet  per  second  per  second.  Find  the  ten- 
sion in  the  rope. 

478.  An  elevator,  starting  from  rest,  has  a  down- 
ward acceleration  of  ^  ^  for  i  second,  then  moves 
uniformly  for  2  seconds,  then  has  an  upward  acceler- 
ation of  ^  ^  until  it  comes  to  rest,  {a)  How  far  does 
it  descend  }  {b)  A  person  whose  weight  is  140  pounds 
experiences  what  pressure  from  the  elevator  during 
each  of  the  three  periods  of  its  motion  .? 

479.  A  weight  of  10  pounds  rests  6  feet  from  the 
edge  of  a  smooth  horizontal  table  that  is  3  feet  high. 
A  string  7  feet  long  passes  over  a  smooth  pulley  at 
the  edge  of  the  table  and  connects  with  a  lo-pound 
weight.  If  this  second  weight  is  allowed  to  fall  in 
what  time  will  it  cause  the  first  weight  to  reach  the 
edge  of  the  table  1 

480.  A  body  is  projected  with  a  velocity  of  50  feet 
per  second  in  a  direction  inclined  40°  upward  from 
the  horizontal.     Determine  the  magnitude  and  direc- 


1  3  4  MECIIANJCS-PR  OBLEMS. 

tion  of  the  velocity  at  the  end  of  2  seconds  {g  being 
taken  equal  to  32.15). 

Let  ACE  be  the  path  of  the  projectile.  The  vertical  velocity 
which  the  body  possessed  when  it  started  from  A  carried  it  to  the 
summit  C  of  the  trajectory,  where  it  had  zero  vertical  velocity,  and 
when  it  reached  E  it  would  possess  its  initial  velocity,  which  would 
be  «  X  sin  a.  (In  Prob.  480  a  is  40°.)  The  constant  horizontal 
velocity  would  be  u  x  cos  a. 

The  vertical  velocity  acquired  in  falling  from  the  highest  point  to 
the  horizontal  AE  would  be  g  x  /, 

.*.  ^  X  /  =  ?/  X  sin  « 
and  the  time  from  A  to  highest  point 

_  ?/  X  sin  a 

g       . 
and  the  total  time  of  flight 

2  ?/  X  sin  a 


The  range  AE 


=  the  horizontal  component  of  veloc- 
ity X  the  time  of  flight 
2  u  sin  a 


ti  X  cos  a  X 
?/2  sin  2  a 


g 


A  g 

The  above  explanation  and  formulas  will  be  of  material  assist- 
ance in  solving  the  problems  that  follow.  In  all  of  these  problems 
the  resistances  of  the  atmosphere  are  neglected. 

481.  A  bullet  is  fired  with  a  velocity  of  i  000  feet 
per  second.  What  must  be  the  angle  of  inclination, 
in  order  that  it  may  strike  a  point  in  the  same  horizon- 
tal plane,  at  a  distance  of  15  625  feet } 

482.  From  the  top  of  a  tower  a  stone  is  thrown  up 
at  an  angle  of  30°,  and  with  a  velocity  of  288  feet  per 
second  ;  the  height  of  the  tower  is  160  feet.  Find 
the  time  required  for  the  stone  to  reach  the  ground, 
and  the  distance  it  will  be  from  the  tower. 


MOTION. 


125 


483.  From  a  train  moving  at  60  miles  per  hour  a 
stone  is  dropped  ;  the  stone  starts  at  a  height  of  8 
feet  above  the  ground.  Through  what  horizontal 
distance  will  the  stone  go  while  falling  t 

484.  A  stone  from  a  quarry  blast  has  a  velocity  of 
200  feet  per  second,  in  a  direction  inclined  at  an 
angle  of  60°  to  the  horizontal  plane.  To  what  height 
will  it  rise,  and  how  far  away  will  it  strike  the  ground  .'* 

485.  A  bullet  is  fired  with  a  velocity  of  which  the 
horizontal  and  vertical  components  are  80  and  120 
feet  per  second  respectively.  Find  the  range  and 
greatest  height. 

486.  The  top  of  a  fortification  wall  is  50  feet  above 
the  level  of  a  city.  From  a  man-of-war  in  the  bay 
300  feet  below  the  top  of  the  wall  and  distant  hori- 
zontally 3  000  feet,  a  projectile  is  fired  with  velocity 
of  I  000  feet  per  second.  The  projectile  just  clears 
the  wall.     Where  will  it  land  inside  the  city  } 


(/ 


3jn 


000  X  /  X  sin  rz  —  ^  gi-  =  300 
I  000  X  /  X  cos  a  =  3000 


Eliminate  /  and  solve  for  a  (a  =  S°  28'). 


1 36  MECHANICS-rROBLEMS. 

Find  the  greatest  height  h,  then  d  being  known  will  enable  one 
to  find  the  time  for  the  projectile  to  fall  that  height  or  to  pass  hori- 
zontally over  the  distance  /.  To  /  add  half  the  range  and  thus  find 
the  distance  from  man-of-war  to  where  the  projectile  will  land  inside 
the  city. 

487.  A  ball  is  discharged  with  an  initial  velocity 
of  I  100  feet  per  second.  How  many  miles  is  the 
greatest  possible  range  1 

488.  A  cannon  ball  is  fired  directly  from  a  hill 
that  is  on  the  coast  and  900  feet  high  :  find  the  time 
which  elapses  before  it  strikes  the  sea. 

489.  A  projectile  is  fired  horizontally  from  the  top 
of  a  hill  300  feet  high  to  a  ship  at  sea.  Its  initial 
velocity  is  2  000  feet  per  second  and  its  weight  500 
pounds.  What  will  be  its  range,  and  what  will  be  the 
energy  of  the  blow  that  it  strikes } 

490.  What  velocity  must  be  given  to  a  golf  ball  to 
enable  it  just  to  clear  the  top  of  a  fence  at  12  feet 
higher  elevation  and  100  yards  distant,  if  the  ball  is 
struck  upv/ards  at  an  angle  of  45° .? 

491.  The  explosive  force  of  a  shell  is  to  be  regulated 
by  proper  charging  so  that  a  required  velocity  can  be 
attained.  Find  what  velocity  will  be  required  for 
it  just  to  clear  a  fortification  wall  the  top  of  which 
is  distant  horizontally  i  mile  and  at  elevation  300  feet 
above  the  gun.     Angle  of  projection  is  45°. 

492.  A  rifle  projects  its  shot  horizontally  with  a 
velocity  of  i  000  feet  per  second  ;  the  shot  strikes  the 


MO  TION. 


137 


OF  THE     *^ 

UNIVERSITY 

OF 


ground  at  a  distance  of   i  000  yards.     What  is  the 
height  of  the  rifle  above  the  ground  ? 

493.  What  is  the  pres- 
sure exerted  horizontally 
on  the  rails  of  an  engine 
of  20  tons  weight  going 
round  a  level  curve  of  600 
yards  radius  at  30  miles 
an  hour  ? 

Centrifugal^  W   7^^ 
force  g    r 

Fig.  72. 
To  derive  the  above  formula  : 

Let  A  and  B  be  two  positions  of  the  engine.     At  A  the  velocity, 

which  is  30  miles  an  hour,  would  be  in  the  direction  of  tangent  v, 

and  at  B  the  same  velocity  would  be  in  direction  of  tangent  v.     In 

going  from  A  to  B  the  direction  of  the  velocity  has  changed,  and  the  • 

measure  of  this  change  is  the  centrifugal  force. 

Its  value  depends  upon  the  rate  of  change  of  motion.     To  find 

the  rate,  from  A  draw  AD  to  represent  the  velocity  of  position  B. 

Then  CD  will  represent  the  velocity  of   B  relative  to  A,  and  its 

v 
value  will  be  AB  x  -,  as  found  from    similar  triangles  ACD    and 

ABO  ;  and  AB  =  velocity  (on  the  curve)  X  time,  so  that  CD,  the 

V 

velocity  of  B  relative  to  A  =  7'/  x  -,  and  the  rate  of  change  a  = 

vt  Y.  -  -=r  t  =  -.     Thus  knowing  the  rate  of  change  or  acceleration 
r  r 

a  the  centrifugal  force  c  can  be  found. 

tr  :  W  =  a  \  g 

~  g     r 

494.  A  train  of  60  tons  weight  is  rounding  a  curve 
of  radius  one  mile,  with  a  velocity  of  20  miles  an 
hour.     What  is  the  horizontal  pressure  on  the  rails  ?         V 


1 38  MECHANICS-PROBLEMS. 

495.  A  24-ton  engine  is  rounding  a  curve  of  400 
yards  radius ;  the  horizontal  pressure  on  the  rails  is 
4.84  tons.     What  is  the  velocity  of  the  engine  ? 

496.  The  rim  of  a  pulley  has  a  mean  radius  of  20 
inches  ;  its  section  is  6  inches  broad  and  \  inch  thick. 
It  revolves  at  200  revolutions  per  minute.  What  is 
the  centrifugal  force  per  inch  length  of  rim  1 

497.  The  mass  of  the  bob  of  a  conical  pendulum  is 
2  pounds,  the  length  of  the  string  is  3  feet,  the  angle 
of  inclination  to  vertical  is  45°.     What  is  the  tension  } 

The  three  forces  acting  on  the  bob  are  :  its  weight  downward,  the 
tension  in  the  string,  and  the  centrifugal  force  outward. 

498.  The  mass  of  the  bob  is  20  pounds,  the  length 
of  the  string  is  2  feet,  the  tension  of  the  string  is 
5007r^  pounds  weight.  How  many  revolutions  per 
second  is  the  pendulum  making  t 

499.  If  a  conical  pendulum  be  10  feet  long,  the 
half  angle  of  the  cone  30°,  and  the  mass  of  the  bob 
12  pounds,  find  the  tension  of  the  thread  and  the 
time  of  one  revolution. 

500.  A  ball  is  hung  by  a  string  in  a  passenger  car 
which  is  rounding  a  curve  of  i  000  feet  radius,  with 
a  velocity  of  40  miles  an  hour.  Find  the  inclination 
of  the  string  to  the  vertical. 

501.  A  ball  is  hanging  from  the  roof  of  a  railroad 
car.  How  much  will  it  be  deflected  from  the  vertical 
when  the  train  is  rounding  a  curve  of  300  yards 
radius  at  speed  of  45  miles  an  hour } 


MOTION. 


139 


502.  Find  the  speed  at  which  a  simple  Watt  Gov- 
ernor runs  when  the  arm  makes  an  angle  of  30°  with 
the  vertical.  Length  of  arm  from  center  of  pin  to 
center  of  ball,  18  inches.     (Fig.  y^.) 


Fig.  73. 

503.  Find  the  speed  of  a  cross-arm  governor  when 
the  arms  make  an  angle  of  30°  with  the  vertical.  The 
length  of  the  arms  from  center  of  pin  to  center  of 
ball  is  29  inches  ;  the  points  of  suspension  are  7 
inches  apart.     (Fig.  74.) 

504.  The  rotating  balls  on  a  centrifugal  governor 
make  160  revolutions  per  minute  ;  the  distance  from 
the  center  of  each  ball  to  the  center  of  the  shaft  is 
4.5  inches.  The  balls  are  of  cast  iron  and  2\  inches 
in  diameter.  Find  the  centrifugal  force  of  the  gov- 
ernor. 

505.  Find  the  speed  in  revolutions  per  minute  of 
a  cross-arm  governor  when  the  arms  make  an  angle 
of  30°  with  the  vertical,  the  length  of  the  arms  from 
center  of  pin  to  center  of  ball  being  24  inches,  and 
the  points  of  suspension  being  6  inches  apart. 


140  MECHANICS-PROBLEMS. 

506.  Find  the  tension  in  each  spoke  of  a  six-spoked 
flywheel,  8  feet  in  diameter  and  weighing  1344  pounds 
when  making  200  revolutions  per  minute  assuming  all 
its  mass  collected  at  its  rim,  and  that  by  reason  of 
cracks  in  the  rim,  the  spokes  have  to  bear  the  whole 
of  the  strain. 

507.  The  flywheel,  which  burst  in  the  Cambria  Steel 
Company's  mill  Jan.  21,  1904  killing  three  men  and 
seriously  injuring  nine  more,  is  reported  to  have 
weighed  50  tons,  and  to  have  been  about  20  feet 
in  diameter.  If  rim  weighed  35  tons  and  its  weight 
was  acting  at  a  mean  diameter  of  19  feet  what  centri- 
fugal force  did  each  of  the  8  spokes  and  its  portion  of 
the  rim  have  to  withstand  when  the  wheel  was 
"racing"  at  150  revolutions  per  minute.-* 

508.  A  man  standing  on  a  street  corner  was  injured 
by  a  horse-shoe  that  was  thrown  from  the  front  wheel 
of  a  passing  automobile.  The  rubber  tire  caught  up 
the  horse-shoe  by  a  protruding  nail  and  carried  it 
around  to  the  top  of  the  wheel  when  it  was  thrown  off 
with  full  force.  Diameter  of  wheel  was  30  inches,  and 
speed  of  automobile  was  20  miles  an  hour.  What 
velocity  would  the  rim  of  the  wheel  thus  give  to  the 
horse-shoe  t 

509.  The  automobile  of  the  above  problem  was  turn- 
ing the  corner  in  a  curve  of  100  feet  radius,  the  road 
being  level.  Therefore  what  two  centrifugal  forces 
were  acting  at  the  instant  the  i -pound  horse-shoe  was 
at  the  top  of  the  wheel  t     What  were  their  values  t 


V 


MOTION.  141 

510.  If  the  horse-shoe  was  thrown  with  the  full 
velocity  of  the  rim  and  horizontally  from  the  top  of  it 
how  far  away  would  it  land  on  level  ground  ? 

511.  As  the  man  was  standing  6  feet  from  the  track 
of  the  automobile  how  far  from  him  must  have  been 
the  wheel  when  the  horse-shoe  was  thrown  off,  pro- 
vided it  was  thrown  tangentially  to  track  ? 

512.  A  locomotive  that  weighs  35  tons  runs  at  40 
miles  an  hour  on  a  level  grade  round  a  curve  of  3  300 
feet  radius  (about  1°  44').  What  centrifugal  force  is 
produced  ?  What  should  be  the  elevation  of  the  outer 
rail  for  a  standard  gage  track  of  4  feet  lof  inches  ? 

513.  In  the  case  of  problem  512  the  railroad  con- 
template putting  on  a  60-ton  locomotive  and  run- 
ning at  maximum  speed  of  60  miles  an  hour.  What 
lateral  pressure  will  the  spikes  of  the  rails,  if  not 
changed,  then  have  to  withstand  ? 

514.  A  stone  weighing  four  ounces  is  whirled 
around  the  head  90  times  a  minute.  If  the  sling  is 
3  feet  6  inches  long  what  will  be  the  pull  in  it .? 

515.  Given    /  the    length    of  a   simple  pendulum, 

'^X  -  the  time  of  an  oscillation  :    show  how  to  find 

^  g 
approximately    the    height    of    a    mountain    when    a 

seconds  pendulum,  by  being  taken  from  sea  level  to  its 

summit,  loses  n  beats  in  24  hours.     If  n  =  15,  what 

is  the  height  of  the  mountain,  the  radius  of  the  earth 

being  4  000  miles  ? 


\y 


1 4  2  ME  CHANICS-PROBLEMS. 

516.  At  sea-level  a  pendulum  beats  seconds.  At 
the  top  of  a  mountain  it  beats  %6  360  times  in  24 
hours.     What  is  the  height  of  the  mountain  t 

517.  A  pendulum  of  length  156.556  inches  oscil- 
lates in  two  seconds  at  London.  What  is  the  value 
of  p;} 

518.  An  800-pound  shot  is  fired  from  an  81 -ton 
gun,  with  a  muzzle  velocity  of  i  400  per  second :  a 
steady  resistance  of  9  tons  begins  to  act  immediately 
after  the  explosion.     How  far  will  the  gun  move  1 

An  impulsive  force  is  a  very  large  force  that  acts  on  a  body  for  so 
short  an  interval  of  time  that  the  body  has  practically  no  motion,  but 
receives  a  change  of  momentum  ;  and  this  change  of  momentum 
measures  the  Impulse  or  effect  produced  by  the  Impulsive  Force. 

In  the  above  problem  the  impulsive  force,  or  action  on  the  shot 
to  drive  it  forward,  is  equal  to  the  reaction  on  the  gun  to  drive  it 
backward. 

Action  =  reaction 
Momentum  before  =  rhomentum  after 
Momentum  of  gun  backward  =  momentum  of  shot  forw'ard 
W  W 

g        s 

and  this  simple  formula,  wath  a  knowledge  of  the  piinciples  of  work, 
will  solve  many  problems  that  involve  questions  of  momentum  of 
two  or  more  bodies. 

For  the  above  problem  :  "^ 

2W0  X  I  400  =  81  X  ?? 

V  =  Yt  ^^^^  P^^"  second,  velocity  of  gun 

at  beginning  of  its  motion. 
s  =  average  velocity  X  time. 

To  find  the  time  :  Motion  has  been  retarded  by  a  force  of  9  tons 
and  the  weight  thus  retarded  is  8r  tons.  Find  a  the  rate  of  retarda- 
tion ;  then  since  the  velocity  of  Yr  equals  a  y^t,  t  can  be  found  and 
lastly  the  space. 


MOTION.  143 

519.  A  56-pound  ball  is  projected  with  a  velocity 

of  I  000  feet  per  second  from  an  8-ton  gun.     What    ^ 
is  the  maximum  velocity  of  recoil  of  the  gun  ? 

520.  A  one-ounce  bullet  fired  out  of  a  20-pound 
rifle  pressed  against  a  mass  of  180  pounds,  kicks  the 
latter  back  with  an  initial  velocity  of  6  inches  per 
second.     Find  the  initial  velocity  of  the  bullet. 

521.  A  shell  bursts  into  two  pieces  that  weigh   12 
pounds  and  20.     The   former   continues    on  with   a 
velocity  of  700  feet  per  second,  and  the  latter  with     (^ 
a  velocity  of  380  feet  per  second.     What  was   the 
velocity  of  the  shell  when  the  explosion  occurred  ? 

522.  A  man  weighing  160  pounds  jumps  with  a 
velocity  of  \6\  feet  per  second  into  a  boat  weighing 
100  pounds.    With  what  velocity  will  boat  move  away  ? 

523.  A  freight  train  weighing  200  tons,  and  travel- 
ing 20  miles  per  hour,  runs  into  a  passenger-train  of 
50  tons  standing  on  the  same  track.  Find  the  ve- 
locity at  which  the  broken  cars  of  the  passenger  train 
will  be  forced  along  the  track,  supposing  e  —  \. 

Momentum  before  =  momentum  after. 
Now  with  this  formula  combine  a  second  law,  namely: 
The  differences  in  velocities  before  x  some  constant  =  the  differ- 
ence in  velocities  after. 

w       w    ,     w        w  ^ 

1.  —  u  ■\ u  —  —  V  -\ V 

g  g  g  g 

2.  (7/  —  u)e  =  V  —  t'' 

(u  and  ;/'  are  velocities  before  impact ;  v  and  ?/'  after  impact.) 

Solving  these  equations,  and 

_  Ww  +  W'//  -  ^W  (u  -  7/') 
^'  ~  W  -f  w 

_  -Wu  +  Wu'  +  W<f  (u  -  1/) 

w  +  w 


1 44  ME  CHANICS-PROBLEMS. 

524.  A  freight  car  of  20  tons  weight  is  switched 
on  to  a  siding  with  velocity  of  i  5  miles  an  hour,  and 
collides  with  another  car  of  10  tons  weight  that  is 
moving  in  the  same  direction  at  5  miles  an  hour.  If 
the  coefficient  of  impact  is  J,  find  the  velocities  of 
the  cars  after  they  collide. 

525.  A  ball  of  mass  4  pounds  and  velocity  4  feet 
per  second  meets  directly  a  ball  of  mass  5  pounds 
with  opposite  velocity  of  2  feet  per  second ;  e  =  \. 
Find  the  velocities  after  impact. 

526.  An  8 -pound  bowling  ball  going  12  feet  a 
second  overtakes  and  strikes  directly  a  lo-pound  ball 
going  6  feet  a  second.  Find  their  velocities  after 
striking  when  the  coefficient  of  impact  e  is  |. 

527.  A  body  weighing  10  pounds,  and  moving 
at  the  rate  of  1 5  feet  a  second,  strikes  another  body 
B  weighing  20  pounds,  and  moving  at  the  rate  of  10 
feet  a  second,  in  the  direction  at  right  angles  to  that 
of  A's  motion.  The  bodies  are  to  be  treated  as  points, 
and  the  impact  is  supposed  to  take  place  in  the  direc- 
tion of  A's  motion.  Find  the  velocities  and  directions 
of  the  motions  of  the  bodies  after  impact,  the  restitu- 
tion being  perfect  (coefficient  of  elasticity  =  i). 

Plot  a  figure  to  show  the  conditions.  The- velocity  of  each  in  a 
direction  perpendicular  to  a  line  through  their  centers  is  unchanged 
by  the  impact. 

Note  that  u  of  formula  for  problem  523  becomes  tt  X  cos  45° ; 
u'  becomes  vf  X  cos  90°;  v,  v  cos  180°;  and  v\  7/  cos  45°. 


REVIEW.  145 


IV.    REVIEW 

Many  of  the  review  problems  that  follow  have 
been  prepared  from  actual  engineering  conditions. 
They  are  classified  somewhat  according  to  their  sub- 
ject matter,  but  are  not  given  in  graded  order,  nor 
with  solutions,  as  they  are  intended  especially  for 
students  who  have  already  pursued  a  course  in 
Mechanics,  or  for  engineers  in  practice  who  may 
wish  to  "brush  up  a  bit." 

528.  One  of  the  largest  chimneys  in  America  is  that 
of  the  Clark  Thread  Co.  at  Newark,  N.J.  Its  height 
is  335  feet,  interior  diameter  1 1  feet,  outside  diameter 
at  base  28^  feet,  at  top  14  feet.  Find  the  work  done 
in  raising  the  material  from  the  ground  to  its  place  in 
the  chimney. 

The  volume  may  be  determined  by  considering  the  whole  cone 
and  subtracting  the  top  and  the  core.  The  average  height  to  raise 
the  material  is  found  to  be  1 19.07  feet ;  the  average  weight  of  mate- 
rial is  130  pounds  per  cubic  foot. 

529.  By  tests  at  the  U.S.  Naval  Academy  a  concrete  pile  of 
conical  shape  19  feet  long  and  tapering  from  6  inches  at  the' point  to 
20  inches  at  the  head,  was  driven  \  inches  by  2  blows  from  a  2  100- 
pound  hammer  falling  20  feet.  The  same  hammer  and  fall  by 
two  blows  drove  a  wooden  pile  also  19  feet  long,  but  9^  inches  at  the 
point  and  11  inches  at  the  head,  a  distance  of  5^5^  inches.  The  re- 
port says  :  "  This  shows  the  comparative  bearing  power." 

What  were  the  resistances  {a)  of  the  concrete  pile 
{b)  of  the  wooden  } 


146 


MKCHANICS-PROBLEMS. 


Fig.  75.    Half-way  Home. 

530.  A  60-inch  McCormick  water  turbine  at  the 
Talbot  Mills  in  North  Billerica,  Mass.  was  tested  by 
the  author  and  engineering  students  in  December, 
1905. 

The  quantity  of  water  entering  the  turbine  was, 
by  measurements  with  a  Price  current  meter,  found  to 
be  7  946  cubic  feet  per  minute,  with  speed  gate  open 
40  per  cent ;  net  fall  given  by  difference  in  reading 
of  gauges  in  foreway  and  raceway,  10.7  feet.  How 
many  horse-power  was  the  water  delivering  to  the 
turbine,  that  is,  what  was  the  "  input  .-^ " 

531.  The  power  available  for  manufacturing  pur- 
poses was  measured  by  a  friction  brake.  (See  Fig. 
'j6}j  Length  of  arm  was  12.00  feet,  revolutions  of 
pulley  100  per  minute,  net  reading  on  platform 
scales  for  the  above  test  was  400   pounds.     What 


KI:\'JK\V 


A7 


horse-power  was  developed,  that  is,  what  was  ''the 
output?"  "The  in-put"  by  problem  530  being 
161  horse-power,  what  was  at  that  time  the  efficiency 
of  the  turbine  with  its  set  of  bevel  gears  and  50  feet 
of  horizontal  shafting  ? 


Fig.  76. 

532.  The  strap  underneath  the  pulley  in  Fig.  76  is  tightened 
by  the  nuts  A  and  B  on  top  of  the  lever.  When  the  brake  is  in  use 
one  nut  can  be  tightened  by  a  pull  of  50  pounds  on  a  3-foot  wrench, 
the  other  by  25  pounds. 

Which  nut  should  tighten  harder  and  what  will 
be  the  approximate  tension  in  that  end  of  the  strap, 
the  threads  being  6  to  the  inch  with  mean  diameter 
of  i^  inches,  and  the  hexagonal  nut  having  a  mean 
outside  diameter  of  2 J  inches?  Use  0.25  as  the  co- 
efficient of  friction  between  the  nut  and  its  washer, 
and  0.15  for  the  threads. 

533.  For  a  bridge  like  Fig.  77  with  lower  chord  a 
para])()la  200  feet  span  and  30  feet  rise  what  would  be 
the  stresses  at  the  abutments  and  middle  wlicn  the 
vertical  reaction  at  each  end  is  600  000  pounds  ? 


148 


MECHANICS-PROBLEMS. 


Fig.  77.    The  Driving  Park  Bridge  at  Rochester,  N.  Y. 

534.  The  platform  of  a  suspension  foot-bridge  100 
feet  span,  10  feet  width,  supports  a  load,  including 
its  own  weight,  of  150  pounds  per  square  foot.  The 
two  suspension  cables  have  a  dip  of  20  feet.  Find 
the  force  acting  on  each  cable  close  to  the  tower,  and 
in  the  middle,  assuming  the  cable  to  hang  in  a  para- 
bolic curve. 

535.  Find  analytically  the  stress  in  the  cable  at 
a  horizontal  distance  of  30  feet  from  the  center. 

536.  The  weight  of  a  fly-wheel  is  8  000  pounds  and 
the  diameter,  20  feet  ;  diameter  of  axle,  14  inches  ; 
coefficient  of  friction,  0.2.  If  the  wheel  is  discon- 
nected from  the  engine  when  making  27  revolutions 
per  minute,  find  how  many  revolutions  it  will  make 
before  it  stops. 


V 


REVIEW.  149 

537.  Experiment  shows  that  a  weight  can  lift  only 
three-quarters  of  its  own  weight  by  means  of  a  rope 
over  a  single  pulley,  this  being  on  account  of  the 
stiffness  of  the  rope  and  the  friction  of  the  axis. 
Hence  show  that  the  mechanical  advantage  of  four 
such  pulleys  arranged  in  two  blocks  is  about  2.05. 


Fig.  78.    Concrete  dam  falling  into  place,  Niagara,  N.  Y. 

Height  of  trestle  is  20  feet,  of  dam  50  feet,  cross  section  of  dam 
7  feet,  4  inches  square,  and  base  of  trestle  i  foot  wider  on  the  water 
side.  Tower  was  tipped  by  three  heavy  jacks  until  it  fell,  as  shown 
in  cut.     (See  Eng.  Record  of  Nov.  18,  1905.) 

538.  How  many  feet  out  of  plumb  would  the  top 
of  tower  move  before  starting  to  fall  > 


u 


1 5  O  ME  CIIANICS-1'R  OJU.  EMS. 

539.  What  foot-pounds  of  energy  did  the  falling 
tower  possess  at  the  instant  it  passed  the  level  of  the 
base  ? 

540.  A  bolt  I  \  inches  in  diameter,  9  threads  to  an 
inch,  head  2  inches  outside  diameter,  is  tightened  up 
by  a  wrench  12  inches  long,  and  pull  on  end  of  50 
pounds.  /A  for  threads  is  0.2,  for  nut  0.3.  Find  the 
stress  in  the  bolt. 

541.  In  a  certain  piece  of  street  railway  construction  I  obsei"ved 
that  the  bolts  in  the  fish  plates  were  being  tightened  unusually  hard, 
and  the  workmen  told  me  that  such  bolts  often  broke.  By  test  we 
found  that  a  pull  of  170  pounds  was  being  used  at  a  distance  of  3.75  w 
feet  from  center  of  nut.  There  were  9  threads  to  the  inch,  mean 
diameter  0.86  inches,  average  outside  diameter  of  nut  1.4  inches, 
diameter  of  inside  bearing  0.89    inches.     The  coefficient  of  friction 

for  the  threads  was  about  0.2  and  for  the  nut  0.3. 

Find  stress  in  bolt,  and  then  the  stress  per 
square  inch  at  root  of  thread  which  was  0.60  inches 
in  diameter. 

542.  The  mean  diameter  of  the  threads  of  a  J 
inch  bolt  is  0.45  inches,  the  slope  of  the  thread  .07, 
the  mean  circumference  of  nut  0.38  inches  and  the 
coefficient  of  friction  0.16.  Find  the  tension  in  the 
bolt  when  tightened  up  by  a  force  of  20  pounds  on 
the  end  of  a  wrench  1 2  inches  long. 

543.  The  floating  cantilever  crane  shown  in  the 
illustration  is  hoisting  a  90-ton  turret  off  the  U.  S. 
Monitor  "Florida."  Distance  from  middle  of  crane 
to  the  legs  is  45  feet;  from  legs  to  turret  15  feet. 
Vertical  height  of  legs  60  feet,  distance  aj^art  at  top 


REVIEW. 


151 


10  feet,  at  bottom  50  feet.  Find  the  stresses  for  this 
case  in  the  legs,  and  in  the  inclined  members  at 
center  considering  a  joint  at  that  point. 

544.  The  load  of  90  tons  is  supported  by  two  large 
steel  blocks.  Each  block  has  four  4-foot  steel  sheaves 
thus  using  8  strands  of  \\  inch  cast-steel  rope.     The 


Fig-  79. 

speed  of  vertical  travel  is  50  feet  per  minute,  {a)  What 
will  be  the  velocity  of  travel  of  the  hoisting  strand  ? 
{b)  What  will  be  the  stress  per  square  inch  in  each 
rope  of  each  block } 

545,    Launching  data  for  nine  of  our  recent  warships  was  given 
in  papers  read  before  the  Society  of  Naval  Architects  and  Marine    \, 


152  ME  CIIA  NICS-PR  OBLEMS. 

Engineers  at  New  York,  November,  1904,  and  published  in  abstract 
by  Engineering  News,  Dec,  22,  1904. 

For  the  "  California,"  built  by  the  Union  Iron  Works  of  San 
Francisco,  and  launched  April  28, 1904  : 

Total  moving  weight,  ship,  cradle,  etc.    ...     6  062  tons 

Mean  slope  of  upper  end  of  ways i  in  27.6 

Ship  started  ''  very  slowly."    It  moved  the  first  foot  in  1 1  seconds. 

From  the  above  the  initial  coefficient  of  friction  has 
been  computed  as  .036.      Check  this  result. 

546.  Total  travel  of  the  ship,  in  preceding  prob- 
lem, was  'j'^6  feet;  total  time,  i  minute  16  seconds  ; 
Travel  to  point  of  maximum  velocity,  320  feet ;  time 
to  maximum  velocity,  41.5  seconds;  amount  of  maxi- 
mum velocity,  22.1  feet  per  second.  The  weight  of 
ship  exclusive  of  cradle  was  5  980  tons.  Means  for 
checking  were  rope  stops  (72  were  broken  of  total 
strength  30  tons),  anchors,  and  mud  banks.  Find  the 
average  resistance  that  the  means  of  checking  must 
have  afforded. 

547.  Consult  the  reference  of  problem  545  (Eng. 
News,  Dec.  22,  1904),  and  copy  Fig.  6  in  note-book. 
Explain  why  the  velocity  and  distance  curves  start 
horizontally  from  the  origirf.  In  the  acceleration 
curve  what  was  the  cause  of  the  rise  between  15 
and  30  seconds  of  elapsed  time  .? 

548.  From  curve  of  problem  547  what  was  the 
velocity  at  time  of  41.5  seconds  }  The  acceleration  } 
What  was  the  acceleration  at  5  5  seconds  }  From  that 
data  compute  the  velocity  for  5  5  seconds. 


REVIEW, 


53 


549.  Coal  from  a  barge  (Figs.  80  to  82)  is  hoisted 
to  a  steeple  tower  where  it  is  run  into  a  car  and  by  the 
action  of  gravity  alone  the  car  goes  down,  a  grade 
294  feet  long  in  24  seconds.  It  strikes  a  cross-bar, 
or  "  stopper  "  which  is  pushed  back  a  distance  of  30 
feet  while  the  car  empties  and  for  an  instant  comes  to 
rest.  The  weight  of  the  car  is  2  000  pounds  and  of 
the  coal  4  000.  If  the  car  empties  uniformly  during 
the  30  feet,  what  is  the  average  force  of  resistance 
that  the  cross-bar  exerts  ? 


Fig  .82. 


550.  The  method  of  stopping  the  car  of  problem  549  may  be 
understood  by  referring  to  Figs.  80  to  82.  The  car,  going  down 
the  grade,  picks  up  the  cross-bar  C,  which  is  clamped  to  the  wire 
cable  AB.    As  the  cross-bar  is  pushed  along  the  cable  moves,  and  the 


154  ^^^'  CIIA  NICS-PR  OBL  EMS. 

pulley  E,  around  which  the  cable  passes,  as  shown  in  the  enlarged 
sketch  E",  goes  from  E,  its  initial  position,  to  E',  its  final  position. 
The  travel  of  this  pulley  raises  a  triangular  frame,  that  is  partly  filled 
with  broken  stone,  from  the  position  shown  dotted  to  that  shown  by 
full  lines  in  Eig.  82.  The  mass  of  stone  is  5.5  feet  X  4.8,  as  shown, 
and  I  ^  feet  thick.  The  space  is  one-third  voids ;  weight  of  stone, 
150  pounds  per  cubic  foot  ;  weight  of  wooden  frame,  i  000  pounds. 

Show  that  the  force  exerted  parallel  to  the  cable 
is,  for  the  initial  position  with  pulley  at  E,  about 
220  pounds;  for  the  final  position  E'  about  1925 
pounds. 

551.  When  the  cross-bar  and  wire  cable  of  the  pre- 
ceding problems  move  through  a  distance  of  30  feet 
the  travelling  pulley  E  goes  from  E  to  E'  as  described. 
The  diameter  of  pulley  being  i^  feet  what  will  be  the 
distance  E  E'  t  What  would  be  the  distance  if  pulley 
were  2  feet  in  diameter  t 

552.  One-fourth  of  the  energy  possessed  by  the 
mass  of  stone  when  in  the  final  position  E',  Fig.  82,  is 
lost  in  friction,  and  three-fourths  of  it  is  utilized  to 
"kick"  automatically  the  empty  car  up  the  incline 
from  C  back  to  its  starting  point  A.  If  this  energy  is 
expended  on  the  car  in  a  return  distance  of  30  feet 
what  will  be  the  maximum  velocity  of  the  car  as  it 
starts  back  1 

553.  When  the  bridge  of  Fig.  83  carries  a  crowd 
of  people  making  a  load  of  150  pounds  per  square  foot 
what  will  be  the  reactions  for  the  middle  truss  and 
the  stresses  in  the  inclined  and  horizontal  members 
at  the  abutments  ? 


REVIEW. 


155 


A  modern  highway  bridge  over  the  main  tracks  of  the  Reading 
raihoad  at  Seventeenth  and  Indiana  Streets,  Philadelphia.  Three 
Pratt  trusses,  with  24-feet  clear  roadways  and  two  lo-foot  sidewalks 
outside.  Concrete  abutments.  The  middle  tru.ss  is  135  feet  of  inches 
from  center  to  center  of  end  pins,  and  22  feet  from  ienter  to  center 
of  chords.     Equal  panels. 


156  MECIIANICS-PJWBLEMS. 

554.  A  bridge  of  the  type  shown  in  Fig.  83  is  sued 
for  a  double-track  raih'oad.  Length  of  bridge  is  150 
feet,  there  are  6  equal  panels,  and  height  of  trusses 
is  25  feet.  With  loading  of  Fig.  85  and  the  second 
driver  of  forward  locomotive  placed  at  the  first  panel 
point  from  abutment,  what  will  be  the  stresses  in  the 
inclined  member  and  the  first  panel  of  lower  chord  of 
the  truss  which  is  carrying  two-thirds  of  the  loadings  .? 

555.  Span  1 1  of  the  Benwood  Bridge  on  the  Balti- 
more and  Ohio  Railroad  is  347  feet  long.  It  was 
reconstructed  in  1904  and  designed  to  carry  the 
heavy  loading  shown  in  Fig.  85.  What  would  be  the 
reactions  for  a  train  half-way  across  the  bridge  1 

556.  With  the  forward  truck  just  going  off  the 
bridge,  what  will  be  the  reactions } 

hfitl.  The  locomotive  of  the  Empire  State  Express 
has  four  drivers  and  a  total  weight  of  124  000 
pounds ;  the  weight  on  the  drivers  is  84  000  pounds ; 
the  coefficient  of  friction  between  wheels  and  rails  is 
0.18.  Find  the  total  weight  of  itself  and  train  which 
it  can  draw  up  a  grade  of  i  in  100,  if  the  resistance 
to  motion  is  1 2  pounds  per  ton. 

558.  From  the  following  data  given  by  a  General 
Superintendent,  determine  what  revolutions  per  minute 
the  locomotive  drivers  were  making  : 

"  The  driver  wheels  are  42  inches  in  diameter,  each  pair  being 
geared  to  a  200  HP,  625  volt-motor,  with  ratio  of  gearing  8i  to  19, 
providing  for  a  total  tractive  effort  at  full  working  load  on  8  motors 
of  70  000  lbs.  and  at  starting  of  80  000  lbs.,  assuming  25%  tractive 
coefficient,  giving  a  nominal  rating  of  i  600  HP.  The  free  running 
speed  of  these  locomotives  is  about  20  miles  per  hour." 


REVIEW. 


157 


m 

Q 
c 

c 

c 


i   c 


c 


o- 

G- 
C 

C 

A 

a 


C 
C 

c 


ic 


-Tc-  25000 

00 

-*-  60  000 
-■^-  50000 
-y-  50000 

--51-    50000 

o 

— X-   32  500 

o 

-^-    32500 

o 

-^-   32500 
-)c-   32  000 

-^-  26  000 

00 

-X-  60  000 

-4-  50000 

-^-  50000 
10 

-*-  50000 


-^-  32  500 
-"k-   32500 

-^^-    32  500 


-r-  -S- 


Dw«.         AxU 

in.  ft.        Load 

in  lb: 


C-r"    ^*-    32000 


Q  2 

-!"  00  O 

o  a  {H 

K  2  W 

P  o  ^ 


h   « 


C  »  oi 
o  o  rt 
Oh^5 


Fig.  84  shows  a  consolidated  locomotive  typical  of  modern  de- 
sign and  development  ;  Fig.  85,  two  such  locomotives  and  their 
train  load  as  used  in  computations  of  modern  railroad  bridges. 


158  MECHANICS-PROBLEMS. 

559.  Also  from  the  foregoing  data  determine  what 
revolutions  the  motors  were  making.  What  amperes 
were  supplied  to  the  8  motors  ? 

560.  An  enormous  freight  locomotive  —  a  Mallet 
duplex  compound  —  designed  as  a  "  mountain  helper  " 
was  put  in  service  on  the  Baltimore  and  Ohio  Railroad 
in  January,  1905.  This  locomotive  has  drawn  36  steel 
cars  weighing  702  tons,  and  i  66^  tons  of  lading,  up 
a  1%  grade,  and  with  an  average  speed  of  10^  miles 
an  hour  ;  weight  of  locomotive  with  tender  and  an 
average  amount  of  coal  and  water  is  225  tons.  What 
horse-power  without  friction  was  developed  for  haul- 
ing the  above  total  load  t 

561.  What  per  cent  of  the  work  done  in  the  preced- 
ing problem  would  be  paying  work  t 

562.  The  draw-bar  pull  of  this  Mallet  Compound 
has  been  found  to  be  74  000  pounds.  When  running 
with  conditions  according  to  problem  560  what  fric- 
tional  resistances  would  exist  .^ 

563.  At  the  testing-plant  of  the  Pennsylvania  Railroad  at  the 
St.  Louis  Exhibition  in  1904  a  freight  locomotive  of  type  two-cylinder 
cross-compound  consolidation  (2-8-0),  size  23  &35  X  32,  made  by 
the  American  Locomotive  Company  for  the  Michigan  Central  Rail- 
road, gave  the  following  data  :  Driving  wheels  5  feet  3  inches  in 
diameter,  total  weight  189000  pounds,  on  drivers  164500;  maxi- 
mum tractive  effort,  sand  being  used  and  locomotive  acting  as  a 
compound,  was  31  838  pounds. 

According  to  the  above  data  what  would  be  the 
coefficient  of  friction  between  drivers  and  rails  .•* 


REVIEW. 


159 


564.  With  speed  of  15.01  miles  per  hour,  80.18 
revolutions  per  minute,  piston  speed  of  428  feet  per 
minute,  the  indicated  horse-power  was  734.9 ;  dyna- 
mometer horse-power  675.7.  Find  the  draw-bar  pull 
and  the  per  cent  of  indicated  horse-power  that  was 
lost  in  friction. 


565.  A  car  is  supported  on  four  36-inch  wheels 
with  4-inch  axles  and  coefficient  of  friction  0.05. 
What  traction  will  be  required  to  move  the  car  on 
a  level  track  with  a  total  weight  of  20  ton  on  the 
axles  .-*  What  energy  will  be  lost  in  friction  per 
minute  with  the  car  moving  30  miles  an  hour } 


iin 

S  S  ^ 

i  ■ 

Fig.  86. 

A  block  of  maple  wood  8  inches  long  and  2x3  inches  in  cross- 
section  is  being  tested  in  a  60  000-pound  Olsen   testing  machine. 


1 60  ME  CHANICS-rROBLEMS. 

The  load  on  the  test  piece  at  the  time  of  failure  is  45  40c  pounds. 
The  plane  of  fracture  is  shown  by  the  white  line  on  the  test-peice, 
Fig.  86.      This  plane  makes  an  angle  of  23°  with  the  horizontal. 

566.  What  would  be  the  pressure  in  the  direction 
of  the  plane  at  'the  time  of  breakage  ?  Find  the 
number  of  square  inches  thus  resisting  this  pressure 
and  then  the  stress  per  square  inch  —  which  is  known 
in  applied  mechanics  as  the  Shear. 

567.  A  horse  is  pulling  a  300-pound  cake  of  ice  up 
a  plank  run  which  makes  an  angle  of  40°  with  the 
horizontal.  There  are  two  single  pulleys  which  have 
efficiencies  of  80%  each.  Coefficient  of  friction  on 
plank  run  0.05.     What  pull  must  horse  exert } 

568.  A  ring  of  weight  W  is  free  to  slide  on  a 
^  smooth  circular  wire  that  stands  in  a 
\  \  vertical  plane.  A  string  attached  to 
^->^lc    the  ring  passes  over  a   smooth  pin  at 

A      the  highest    point  of    the   circle  and 
w     sustains  a  weight  P.     Determine  the 
Fig.  87.  position  of  equilibrium. 

569.  CD  is  a  vertical  wall.  A  is  a  point  of  sup- 
port 12  feet  from  the  wall.  ED  is  a  uniform  bar  32 
feet  long  resting  on  A  and  against  the  wall  CD.  All 
the  surfaces  are  smooth.  Find  the  position  of  equi- 
librium of  the  bar. 

570.  Fig.  88  shows  a  Carson-Lidgerwood  cable  way  in  use  at 
Hartford,  Conn.,  lowering  a  12-foot  length  of  36-inch  cast-iron  pipe. 
The  pipe  is  part  of  an  intercepting  sewer  that  passes  under  a 
river  by  means  of  an  inverted  siphon.  Weight  of  pipe  is  3  tons, 
span  of  cableway  300  feet,  and  the  design  of  cableway  is  such  that, 


/ 


REVIEW. 


6i 


Fig.  88. 

as  customary,  the  sag  by  a  full  load  in  the  middle  is  allowed  to  be 
^Q  of  the  span. 

The  practical  method  used  by  manufacturers  of  trench  machinery 
for  computing  the  total  stress  in  cableways  of  this  sort  is  to  con- 
sider the  condition  of  maximum  loading,  namely  with  the  load  in 
the  middle  of  the  span,  and  for  that  condition  to  divide  half  the 

span  by  double  the  sag  [  ^^ —  )  •     This  gives  a  factor  which  mul- 


OF  THE  A 

UNIVERSITY    \ 

OF 


1 62  MECHANICS-PROBLEMS, 

tiplied  by  the  total  load  (in  this  case  3  tons  +  weight  of  cable)  gives 
the  required  stress  in  the  main  cable- 

How  does  the  result  obtained  by  the  above  prac- 
tical method  check  with  result  obtained  by  the 
method  of  parallelogram  of  forces,  considering  the 
total  load  of  3  tons  -f-  weight  of  cable  as  acting  at 
middle  of  span  ? 

571.  Each  tower  for  the  above  cableway  is  30  feet  high,  and  con- 
sists of  a  vertical  timber  frame,  or  bent,  that  is  formed  by  two  8  X 
10  inch  legs  spread  9  feet  apart  at  the  bottom  and  2  feet  at  the  top. 
The  main  cable  passes  over  an  iron  saddle  at  the  top  of  the  tower 
and  is  fastened  to  a  "  dead-man  "  anchorage  that  is  buried  60  feet 
from  the  foot  of  the  tower.  The  timber  frame  is  kept  vertical  by 
steel  guys  made  fast  at  top  of  tower  and  to  the  same  "  dead  man." 

What  horizontal  .  stress  do  these  guys  have  to 
provide  for  when  the  main  cable  is  free  to  slip  on  the 
saddle  ?     What  vertical  load  do  they  add  to  the  tower  } 

572.  The  pull  on  the  hoisting  ropes  causes  an 
additional  stress  that  would  be,  for  this  case,  equiva- 
lent to  a  vertical  load  of  about  3  tons  on  one  tower. 
Add  the  three  vertical  stresses  due  to  main  cable, 
steel  guys,  and  hoisting  ropes  and  then  find  the  stress 
in  each  leg  of  the  tower. 

573.  A  12-inch  Pelton  water-motor  of  3  horse- 
power is  tested  by  a  friction  brake  that  encircles 
three  fourths  of  a  4-inch  pulley  on  the  motor  and  has 
a  lever  arm  that  extends  22  inches  from  center  of 
pulley  to  scales.  The  scales  read  5  pounds  when 
motor  is  making  i  1 50  revolutions  per  minute.  What 
horse-power  is  being  developed  t 


REVIEW. 


163 


574.  A  6-inch  water-pipe  that  is  600  feet  long  is 
deHvering  750  gallons  of  water  per  minute  ;  the  water 
is  shut  ojf  by  uniformly  closing  a  6-inch  vaK'e  in  3 
seconds  of  time.  How  much  will  the  static  pressure 
near  the  valve  be  increased  ? 


|< 69  «.  H       ' 


Fig.  89. 

575.  A  water-works  tank  is  on  a  trestle  which 
stands  on  uneven  ground  as  shown  in  diagram. 
The  tank  weighs  30  000  pounds.  A  strong  wind 
gives  a  pressure  of  40  pounds  per  square  foot.  Find 
the  stress  in  the  plane  of  the  legs  DB  (a)  when  the 
tank  is  empty  ;  {b)  when  the  tank  is  full,  {c)  Find 
how  much  water  will  prevent  the  tank  from  over- 
turning. 


V 


The  wind  acting  on  the  curved  surface  of  the  tank  causes  a 
pressure  that  may  be  taken  as  0.6  of  that  on  a  vertical  section 
through  the  middle  of  the  tank. 


164 


ME  CI/ A  NICS-rR  O  BLEMS. 


576.  Name  two  advantages  of  the  hemispherical 
(or  similar)  bottom  over  a  flat  bottom  as  represented 
in  Fig.  89.  A  tank  20  feet  high  on  the  sides  and  20 
feet  in  diameter  with  hemispherical  bottom  will  hold 

how  many  gallons  of 
wat er  ?  Wh  at  wi  1 1  b e 
the  wind  pressure  if 
taken  as  in  preceding 
problem  ? 

577.  A  clause  in  pro- 
posed specifications  for 
water  valves  requires  that 
''  a  valve  shall  stand  with- 
out injury  a  pull  of  175 
pounds  on  a  wrench  that 
is  in  lenglh  \\  times  the 
radius  of  the  wheel."  In 
considering  these  specifica- 
tions an  engineer  and  in- 
spector asks  if  this  test 
unduly  strains  a  valve  ?  The 
following  analysis  can  be 
made  relative  lo  an  S-inch 
outside  screw-and-yoke 
valve :  Diameter  of  hand 
wheel  is  14  inches,  diam- 
eter of  spindle,  if  inches,  4 
threads  to  the  inch  (mean 
bearing  diameter  i^  inches). 
Valve  seats  taper  from  4  inches  to  2  inches  in  diameter  of  valve 
—  8  inches.  Bearing  of  hand  wheel  has  a  mean  diameter  of  2  inches. 
The  coefficient  of  friction  for  the  bearing  of  hand  wheel,  the  threads, 
and  the  face  of  valve  against  its  seat,  may  be  taken  as  0.15. 

Find  the  stress  in  the  spindle  caused  by  the  pull 
of  175  pounds  as  indicated  above. 


Fig.  90.  A  modern  form  of  water-tank. 
(Erected  at  St.  Elmo,  111.,  by  the  Chicago 
Bridge  and  Iron  Works.) 


HE  VIE  IV. 


165 


To  find  this  stress  it  is  only  necessary  to  consider 
conditions  that  affect  the  friction  of  the  hand  wheel. 
The  resistances  at  the  valve  have  no  affect  on  the 
stress  in  the  spindle  which  in  any  case  is  subjected  to 
that  part  of  the  stress  that  is  transmitted  by  the  hand 
wheel.  To  compute  this  stress  consider  one  revolu- 
tion of  the  hand  wheels. 

Work   =   Work     +Work         +  Work 

of  pull  of  lifting  on  threads  on  bearing 

Substitute  and  find  the  unknown  term  W,  the  stress. 


Fig.  9'. 


Fig.  9a. 


i66 


ME  CIIA  NICS-PROBLEMS 


578.  Find  the  pressure  against  the  seat  of  the  valve 

(no  water  pressure  being  considered)  when  a  pull  of      \/ 
i^j^  pounds  is  applied  as  in  problem  577. 

579.  When  a  water  pressure  of    100   pounds  per 
square  inch  acts  on  one  side  of  the  8  inch  valve  in 
problem    578  and    the   test    of    175  pounds   pull  is     1/ 
applied  what  normal  pressure  exists  on  each  side  of 
the  valve  ? 


12/t. 

A  length  of  water  pipe 


Spinot  end 


2.25  in.  -*J 


8  Half  Sectiqn  of  an  18-incli  water  pipe 

Fig.  93. 

Dimensions  and  weights  for  cast-iron  water  pipe  are  given  in  the 
*'  Standard  Specifications  for  Cast-iron  Pipe  and  Special  Castings," 
issued  Sept.  10,  1902,  by  the  New  England  Water  Works  Asso- 
ciation. 


REVIEW.  167 

580.  The  dimensions  for  an  18-inch  pipe  of  class  D 
designed  for  a  hydrostatic  pressure  of  300  pounds  per    / 
square  inch  are  represented   in    Fig.  93.     Find  the 
weight  of  portion  E  D  and  then  the  total  weight  of 
the  whole  length  of  pipe. 

581.  A  roof  has  triangular  trusses  12  feet  apart. 
Weight  of  roof  covering  and  snow  equals  30  pounds 
per  square  foot,  and  the  floor  gives  a  load  equivalent 
to  20  000  pounds  concentrated  at  the  foot  of  a  vertical 
rod  at  the  center  of  the  truss  ;  length  of  truss  is 
40  feet,  height  10  feet.  Find  the  stresses  in  rafters 
and  tie-rod. 

582.  A  triangular  jib-crane  ABC  carries  at  A 
60  000  pounds,  the  line  of  action  being  parallel  to  BC 
which  is  vertical.  AB  =  10  feet,  BC  8  feet,  AC 
1 1  feet.  Find  the  amount  and  kind  of  stresses  acting 
in  AB  and  AC. 

583.  A  derrick  with  mast  40  feet  long  and  boom 
55  feet  long,  set  at  60°  from  the  horizontal,  is  lower- 
ing into  water  a  wrought-iron  pipe  12  feet  long,  60 
inches  internal  diameter,  66  inches  external  diameter.      v 
Density  of  wrought-iron  is   J.Z.     Find   the  stresses 

in  boom  and  tackle  when  the  pipe  is  in  air,  and  also 
when  it  is  in  water. 

584.  Is  the  retaining  wall  shown  in  Fig.  94  safe 
against  overturning  by  the  earth  pressure  acting  as 
represented.?  Does  the  resultant  pressure  between 
the  weight  of  the  masonry,  taken  at   170  pounds  per 


i68 


ME  CHA  NICS-PR  OBLEMS. 


cubic  foot,  and  the  earth  pressure  cut  the  base  *'  within 
the  middle  third  ?" 


I i. 


Fi2.  94- 

585.  Fig.  95  shows  a  retaining  wall  of  masonry  as 
built  at  Northfield,  Vt.  As  in  the  preceding  prob- 
lem, is  the  wall  safe  against  overturning  ?  Where 
does  the  resultant  cut  the  base  ? 

586.  The  waste  gate  of  the  canal  for  the  Nashua 
Manufacturing  Company  at  Nashua,  N.H.,  is  about 
7 J  feet  high  and  4^  feet  wide.  When  this  gate  is 
closed  there  is  usually  a  head  of  10  feet  of  water  on 
its  center.     The  coefficient  of  friction  of  this  wooden 

gate  against  an  ordinary  metal 
seat  is  taken  as  0.40  and  the 
weight  of  the  gate  is  i  000 
pounds.  What  force  in  tons 
\W11  be  required  to  lift  it } 

587.  How  wide  on  top  should 
be  the  dam  shown  in  Fig.  96  to 
withstand   the   reservoir  pres- 


>- 

/ 

Ucservoir 

/    Concrete  Dam 

20 

liO  lbs.  per  cu.  ft. 

1 
1 

Fig.  96. 


sure  with  factor  of  safety  of  8  ? 


REVIEW. 


169 


588.  Two  Indians  wanted  to  divide  a  birch  log, 
that  was  30  feet  long  and  tapering  from  8  inches  in 
diameter  to  1 2,  so  that  each  would  have  one-half.  A 
school  teacher  told  them  to  balance  it  and  saw  it  open 
at  that  point.     At  what    point  should  it  be  cut } 

589.  A  20  pound  shot  is  fired  from  a  2  000-pound 
gun  of  length  10  feet  ;  the  muzzle  velocity  of  the  shot 
being  i  200  feet  per  second  how  far  will  the  gun  \J 
recoil  up  an  incline  rising  i  vertically  to  15  on  the 
slope?  How  long  will  it  take  the  shot  to  travel 
through  the  gun  } 


Fig.  97. 

590.    The  engine  and  geared  drum  shown  in  the  illustration  are 
used  for  hoisting  ore  to  the  top  of  a  blast  furnace.     The  engine  cyl- 


y 


I  7  O  ME  CIIA NICS-PR OBLEMS. 

inder  is  12  inches  in  diameter,  makes  a  15-inch  stroke,  and  300  revo- 
lutions per  minute.  The  mean  effective  pressure  of  the  steam  being 
100  pounds  per  square  inch,  what  horse-power  is  developed  ?  The 
ratio  of  gears  is  5.6,  to  i  ;  the  diameter  of  drum,  \\  feet.  The  effi- 
ciency of  engine,  geared  drum  and  rest  of  mechanism  is  about  85 
per  cent. 

Therefore,  under  the  above  conditions,  what  force 
will  the  engine  give  to  the  cable  for  drawing  the 
loaded  "  skip-car  "  up  the  incline  to  the  top  of  the 
blast-furnace  ? 

591.  The  drum  of  a  hoisting  engine  is  4  feet  in 
diameter.  The  angle  between  the  engine  crank  and 
connecting  rod  is  60°.  Length  of  crank  i  foot,  con- 
necting rod  5  feet.  Steam  pressure  on  the  piston 
100  000  pounds  which  just  balances  a  load  W  that  is 
being  hoisted.  Determine  the  load  W,  the  compres- 
sion in  the  connecting  rod  and  the  side  pressure 
against  the  cross-head  guide. 

592.  Sixteen  horse-power  is  to  be  transmitted  by  a 
belt  which   embraces  |   of  the   circumference   of  a 
20-inch  pulley  that  makes  1 20  revolutions  per  minute  ;        / 
coefficient  of  friction  is  0.35.     Find  {a)  the  tension  in 

the  two  sides  of  the  belt  when  slipping  is  just  pre- 
vented and  {b)  the  width  of  belt  required,  thickness 
being  f  inches,  and  working  stress  300  pounds  per 
square  inch  of  section. 

593.  Find  the  width  of  a  belt  necessary  to  transmit 
10  horse-power  to  a  pulley  12  inches  in  diameter,  so 
that  the  greatest  tension  may  not  exceed  40  pounds 


REVIEW.  171 

per  inch  of  width  when  the  pulley  makes  i  500  revo- 
lutions per  minute,  and  the  coefficient  of  friction  is 
0.25. 

594.  A  test  was  made,  Aug.  15,  1905,  of  the  new 
steam  plant  of  the  Wolff  Milling  Company  at  New 
Haven,  Mo.     The  engine  had  a  high-pressure  cylin-     ^ 
der  1 2  inches  in  diameter  ;  low  pressure,  24  inches ; 
length  of  stroke,  36  inches.     The  revolutions  were 

77.1  per  minute;  the  indicated  horse-power  of  high- 
pressure  cylinder,  85.74,  low  pressure,  66.19.  What 
were  the  mean  effective  pressures  in  the  two  cyl- 
inders } 

595.  The  engine  and  boiler  test  of  problem  594 
was  continued  10  hours.  During  that  time  177.72 
barrels  of  flour  (each  weighing  196  pounds)  had  been 
made,  and  3  685  pounds  of  coal  had  been  burned 
in  the  boilers  ;  cost  of  coal  per  ton  of  2  000  pounds 
was  $2.90.  Find  the  cost  of  coal  for  each  barrel  of 
flour  made,  and  the  pounds  of  coal  burned  per  hour 
per  indicated  horse-power. 

596.  A  Columbus  Gas  Engine  tested  as  shown  by 
Fig.  98,  Oct.  21,  1905,  gave  the  following  data  :  Revo- 
lutions during  15  minutes  3  688,  explosions  i  516,  net  1/ 
load  on  brake  arm  30  pounds,  length  of  arm  5  feet 
3.024  inches.  Four  indicator  cards  taken  during  the 
test  gave  average  areas  0.69  square  inches,  length 
3.00  inches.     Stiffness  of  spring  that  was  used  300. 


1/2 


ME  CI/ A  NICS-PR  OBLEMSS 


The  diameter  of  engine  cylinder  is  'j.'j^  inches  and 
the  length  of  stroke  1 1  inches.      Find  the  efficiency. 

In  computing  the  horse -power  of  a  gas  engine  by  indicator  cards 
the  number  of  explosions  corresponds  to  the  number  of  revolutions 
for  an  ordinary  engine. 


Fig,  98. 

597.  What  would  be  the  indicated  horse-power  of 
the  gas  engine,  shown  on  page  17,  and  which  has  a 
piston  12  inches  in  diameter  and  a  crank  8  inches 
long  ?  The  engine  works  at  1 50  revolutions  a  minute, 
there  is  an  explosion  every  2  revolutions,  and  the 
mean  effective  pressure  in  the  cylinder  is  62  pounds 
per  square  inch. 

598.  The  speed  of  the  governor  shaft  AB  is  500 
revolutions  per  minute.  The  lo-pound  ball  is  to 
be  replaced  by  two   20-pound  balls  that   revolve  in 


REVIEW. 


173 


1 

%^ 

\  \ 

1 

\  \ 

\    \ 

1 

10I6fc 

Fig.  99. 

as 

the    lo-pound 

planes  distant  i  foot  and  4  feet 
from  the  plane  of  the  lo-pound 
ball.  Take  the  distance  AC  as 
7  inches  and  find  R  and  Rj,  the 
distances  at  which  the  20-pound 
balls  will  revolve  from  the  gov- 
ernor shaft  when  their  centrifugal 
forces  have  the  same  moment 
about  the  speed  controller  at  A 
one  alone. 

599.  In  the  preceding  problem  could  the  distance 
AC  be  changed  and  still  have  the  moments  of  the  two 
20-pound  balls  balance  the  10.^  If  so  what  is  one 
such  distance  "^ 

600.  A  rope  manufacturer's  catalogue  states : 

"  The  breaking  strength  of  rope  may  be  taken  as  7  000  x  diam- 
eter squared.  For  a  constant  transmission  the  best  results  are 
obtained  when  the  tension  on  the  driving  side  of  the  rope  is  not 
more  than  ^^^  of  the  breaking  strength  ;  and  the  tension  on  the  driv- 
ing side  is  usually  twice  the  tension  on  the  slack  side." 

Find  the  horse-power  of  a  2-inch  diameter  rope,  of 

weight  per  foot  0.34  X  diameter  squared,  that  runs 

at   3  000  feet  per  minute  when  centrifugal  force  is 

considered. 

Observe  that  the  tension  on  slack  side  =  centrifugal  force  of 
belt  -f  ^  of  difference  in  tension. 


\/ 


1 74  ME  CHANICS-PROBLEMS. 


EXAMINATIONS. 

MECHANICS.* 

YALE  UNIVERSITY,  SHEFFIELD  SCIENTIFIC  SCHOOL. 

Senior  Mechanical  and  Mining  Engineers. 

March,  1905. 

1.  {a)  Define  five  different  units  of  force,  {b)  A 
balloon  is  ascending  with  a  speed  which  is  increasing 
at  the  rate  of  4  feet  per  second  in  each  second.  Find 
the  apparent  weight  of  10  pounds  weighed  by  a  spring 
balance  in  the  balloon. 

2.  A  weight  of  20  pounds  rests  7  feet  from  the 
edge  of  a  smooth  horizontal  table  4  feet  high.  A  string 
8  feet  long  passes  over  a  smooth  pulley  at  edge  of  the 
table  and  connects  with  a  lo-pound  weight.  If  this 
second  weight  is  allowed  to  fall,  in  what  time  will  the 
first  weight  reach  the  edge  of  the  table. 

3.  {a)  A  cord  passing  over  a  smooth  pulley  carries 
10  pounds  at  one  end  and  54  at  the  other  ;  what  will 
be  the  tension  in  the  cord  >  {b)  A  shopkeeper  uses 
a  balance  with  arms  in  ratio  of  5  to  6.  He  weighs 
out  from  alternate  pans  what  appears  to  be  60  pounds. 
How  much  does  he  gain  or  lose  } 

4.  {a)  Define  a  force  couple.  Show  that  a  force 
couple  cannot  be  replaced  by  a  single  force,     {b)  Show 

*  Preparatory  Studies  :  About  20  weeks  of  Mechanics  in  a  three  hour  a  week 
course,  and  the  present  course  of  10  weeks  with  three  hours  a  week  preparation. 


EXAMINA  TIONS.  1 7  5 

how  to  find  the  resultant  of  any  number  of  non-con- 
current forces  acting  on  a  rigid  body. 

5.  {a)  Find  the  force  of  attraction  of  a  homoge- 
neous sphere  on  a  particle  within  the  sphere,  {b)  The 
mass  of  the  sun  is  300  000  times  the  mass  of  the 
earth,  and  its  radius  is  100  times  the  radius  of  the 
earth.  How  far  will  a  stone  fall  from  rest  in  one 
second  at  surface  of  sun  t 

6.  {a)  A  uniform  rod  8  feet  long,  weighing  18 
pounds,  is  fastened  at  one  end  to  a  vertical  wall  by  a 
smooth  hinge.  It  is  kept  horizontal  by  a  string  10 
feet  long,  attached  to  its  free  end  and  to  a  point  in 
the  wall.  Find  the  tension  in  the  string  and  the 
pressure  on  the  hinge,  {b)  A  uniform  rod  AB, 
20  inches  long  weighing  20  pounds,  rests  horizontally 
upon  two  pegs  whose  distance  apart  is  8  inches. 
How  must  the  rod  be  placed  so  that  the  pressure  on 
the  pegs  may  be  equal  when  weights  of  40  and  60 
pounds  are  suspended  from  A  and  B,  respectively } 

7.  Find  by  the  principle  of  virtual  work  the  con- 
dition of  equilibrium  for  a  differential  screw  consider- 
ing friction. 

8.  A  uniform  ladder  70  feet  long  is  equally  inclined 
to  a  vertical  wall  and  the  horizontal  ground.  A  man 
weighing  224  pounds  ascends  the  ladder,  which  weighs 
448  pounds.  How  far  up  the  ladder  can  the  man 
ascend  before  it  slips  if  the  coefficient  of  friction  for 
the  wall  is  \  and  for  the  ground  \  ? 


I  "jG  MECHANICS-PROBLEMS. 

9.  Find  the  work  lost  by  a  shaft  with  a  truncated 
pivot,  bearing  an  end  thrust. 

10.  A  belt  passing  around  a  drum  has  an  angle  of 
contact  a  and  a  coefficient  of  friction  /i.  Find  the 
horse-power  which  can  be  transmitted. 

11.  Two  rough  inclined  planes  are  placed  end  to 
end.  A  body  of  lOO  pounds  rests  on  one  of  the 
planes,  which  has  an  inclination  of  6o°.  A  string 
attached  to  this  body  passes  over  a  smooth  pulley  at 
the  apex  of  the  planes  and  holds  another  body  on  the 
second  plane  of  inclination,  30°.  If  coefficient  of 
friction  for  each  plane  is  ^,  find  the  weight  of  second 
body  to  just  hold  the  first  from  sliding  down  the  plane. 

sine  30°  =  .5  cosine  30°  =  .86 

MECHANICS. 

TUFTS    COLLEGE,   DEPARTMENT    OF     ENGINEERING. 

Examination  at  Mid-Year,  Feb.  6,  1905.* 

Answer  any  eight  questions. 

1.  In  tests  of  cast-iron  fly  wheels  {Eng.  Nezvs,  Dec. 
15,  1904)  record  is  given  of  one  as  follows  :  Diameter 
of  wheel  4  feet,  stress  in  each  arm  due  to  the  centri- 
fugal force  of  its  portion  of  the  rim  1680  pounds, 
weight  of  same  portion  of  rim  yh  pounds.  Find 
bursting  speed  in  miles  per  hour. 

2.  A  leather  belt  treated  with  dressing  has  coeffi- 
cient of  friction  on  an  iron  pulley  of  0.3.     The  belt 

*  Preparatory  studies  :  Physics  lectures  one  year,  laboratory  one-half  year, 
mechanism  one-half  year,  and  present  course  of  half  year  with  three  class  hours  per 
week  and  two  hours  of  preparation  for  each. 


EXAMINA  TIONS.  I  7  / 

encircles  200°  of  a  pulley  10  feet  in  diameter.     When 
running  at  140  revolutions  per  minute  the  belt  must 
transmit  300  horse-power.      How  wide   should  belt 
be  if  it  is  designed  to  stand  100  pounds  per  inch  of" 
width  t 

3.  A  wall  derrick  has  a  vertical  post  9  feet  high, 
at  top  a  horizontal  member  15  feet  long,  and  3  feet 
back  from  the  load  of  10  tons  at  outward  end  is  a 
brace  13  feet  long  connecting  with  the  vertical  post 
at  a  point  4  feet  up  from  ground.      Find  stresses. 

4.  A  highway  bridge  80  feet  long  has  supports 
2  feet  from  one  end  and  10  feet  from  the  other. 
Uniform  load  on  bridge  is  300  pounds  per  linear 
foot.  A  road  roller  of  10  tons  weight  is  half-way 
across  ;  what  load  is  then  on  each  abutment  } 

5.  A  large  type  of  locomotive  recently  put  in  ser- 
vice on  the  N.  Y.  C.  &  H.  R.  R.  has  developed  ap- 
proximately 2  000  horse-power.  How  heavy  a  train 
could  this  locomotive  draw,  at  speed  of  40  miles  an 
hour,  up  a  2  per  cent  grade  —  {a)  without  wind  or 
frictional  resistances,  {b)  with  resistances  of  20  pounds 
per  ton  acting  t 

6.  A  train  of  400  tons  starts  from  a  station  and 'on 
a  level  track  attains  a  speed  of  40  miles  an  hour  in 
one  minute.  Neglecting  resistances,  what  would  be 
the  draw-bar  pull } 

7.  A  stiff-leg  steel  derrick  with  vertical  mast 
55    feet    high,   boom    85    feet  long,  set  with  tackle 


1 7  8  ME  CHA  NICS-PR  OBLEMS. 

40  feet  long  is  raising  two  boilers  of  50  tons  total 
weight.  Find  stresses  in  boom  and  tackle  and  in 
back  stay  which  makes  an  angle  of  30°  with  vertical. 
'If  mast  be  made  of  two  members  joined  at  top  and  20 
feet  apart  at  bottom  what  stresses  must  they  sustain  ? 

8.  What  would  be  the  total  horse-power  of  pumps 
working  12  hours  per  day  to  supply  the  City  of  Med- 
ford,  21  600  population,  with  100  gallons  of  water 
per  day  (for  each  person)  and  forced  against  60  pounds 
pressure  (equals  a  height  of  138  feet)  t  The  efficiency 
of  engines  and  pumps  is  to  be  80  per  cent. 

9.  A  shell  can  be  fired  with  velocity  of  2  000  feet 
per  second  ;  neglecting  resistances,  how  near  to  shore 
can  a  man-of-war  be  in  order  to  have  its  shells  just 
clear  a  fortification  wall  500  feet  above  sea  level,  angle 
of  projection  being  30°  t 

10.  Derive  the  formula  for  centrifugal  force.  The 
20th  Century  express  attains  a  speed  of  60  miles  per 
hour.  When  rounding  a  curve  of  4  000  feet  radius 
how  much  should  the  outer  rail  be  elevated  to  avoid 
lateral  pressure  }  (Center  to  center  of  rails  is  4  feet 
I  of  inches.) 

11.  Define  acceleration,  work,  moment  of  a  force, 
coefficient  of  friction.  Find  the  least  force  necessary 
to  pull  a  packing  case  of  300  pounds  weight  along  a 
horizontal  floor.      Coefficient  of  friction  0.58. 

Total  number  of  problems  taken  during  the  half-year  has  been 
about  195. 


EX  A  MINA  TIONS.  1 79 

MECHANICS. 
TUFTS    COLLEGE,    DEPARTMENT    OF    ENGINEERING. 

Examination  at  Mid-Year,  Feb.  i,  1906* 
Division  a  answer  any  8   questions.     Division  ^  answer  No.  11   and 
7  others. 

1.  In  a  direct-acting  steam  engine  the  piston  pres- 
sure is  22  500  pounds ;  the  connecting-rod  makes  a 
maximum  angle  of  i  5°  with  the  hne  of  action  of  the 
piston.     Find  the  pressure  on  the  guides. 

2.  An  iron  wedge  having  faces  of  equal  taper  that 
make  an  angle  of  10°  is  being  forced  under  an  iron 
column  which  is  supporting  a  load  of  5  tons.  The 
coefficient  of  friction  for  the  iron  surfaces  is  0.18. 
What  force  is  needed  to  push  the  wedge  forward  } 

3.  An  electric  car  that  is  filled  with  passengers 
and  weighs  25  tons  goes  up  a  grade  of  i  in  100  at 
the  speed  of  10  miles  an  hour.  The  total  resistances 
to  traction  are  30  pounds  per  ton.  What  horse-power 
must  be  supplied  when  the  efficiency  of  the  mechan- 
ism is  60  per  cent }  For  an  electro  motive  force  of 
500  volts  what  amperes  would  be  necessary.^ 

4.  A  shaper  head  that  weighs  500  pounds  makes 
its  forward  stroke  of  12  inches  in  6  seconds.  The 
resistances  of  cutting  and  of  machinery  are  equivalent 
to  a  coefficient  of  friction  of  0.5.  At  what  rate  is 
work  being  done  .^ 

*  Preparatory  studies  same  as  for  examination  of  1905  and  given  at  bottom  of 
page  i?6. 


l8o  MECHANICS-PROBLEMS. 

5.  Coal  is  hoisted  from  a  barge  to  a  tower  where  it 
is  run  into  a  car  that  goes  down  a  grade  294  feet  long 
in  24  seconds.  It  strikes  a  cross-bar  or  ''  stopper  " 
which  is  pushed  back  a  distance  of  30  feet  while  the 
car  empties  and  for  an  instant  comes  to  rest.  The 
weight  of  the  car  is  2  000  pounds  and  of  the  coal 
4  000.  If  the  car  empties  uniformly  during  the  30 
feet  what  is  the  average  force  of  resistance  that  the 
cross-bar  exerts } 

6.  A  highway  bridge  of  span  48  feet,  width  40  feet, 
has  two  queen-post  trusses  of  depth  9.2  feet ;  and 
each  truss  is  divided  by  two  posts  into  three  equal 
parts.  The  bridge  is  crowded  with  people  making  a 
load  of  1 50  pounds  per  square  foot,  and  also  an  elec- 
tric car  one-third  the  way  across  the  bridge  causes  an 
additional  load  equivalent  to  a  concentrated  load  of 
20  tons.     Find  the  stresses  in  chords  and  posts. 

7.  The  head  plate  of  a  Buckeye 
engine  is  to  be  hoisted  by  a  con- 
tinuous rope  that  passes  through  eye 
bolts  that  are  5  feet  apart,  and 
through  a  chain-hoist  hook  that  is 
3  feet  above  the  plane  of  the  eye 
bolts.  The  rope  is  free  to  slip,  and 
the  plate  weighs  500  pounds.  Find 
the  total  pull  that  tends  to  break  the  eye  bolts. 

8.  The  center  of  a  steel  crank-pin  that  weighs 
16  pounds  is  12  inches  from  the  center  of  the  engine 


EXAMINA  TIONS,  1 8 1 

shaft.     The  shaft  makes  190  revolutions  per  minute. 
Find  the  centrifugal  force  caused  by  the  pin. 

9.  The  San  Mateo  Dam  in  California  was  designed 
for  a  height  of  170  feet,  width  at  top  25  feet,  at  base 
176  feet,  with  a  uniform  batter  on  the  water  side  4  to 
I,  and  on  the  back  side  near  the  top  2 J  to  i,  then  a 
curve  of  radius  258  feet  to  near  the  bottom  where  the 
batter  is  i  to  i.  The  material  throughout  is  concrete 
of  weight  1 50  pounds  per  cubic  foot.  Compute 
approximately  the  factor  of  safety  of  such  a  section 
against  overturning. 

10.  Define  moment  of  a  force  and  illustrate  by  an 
example.  Also  define  and  illustrate  "resolve  parallel 
and  perpendicular  to  plane,"  a  couple  and  three  other 
important  terms  or  equations  of  Mechanics.  Show 
how  to  find  the  least  force  necessary  to  pull  a  box 
along  a  horizontal  floor. 

11.  ~U  -\ «    =  — V  -\ V 

g  g  g  S 

Z'  —  7<'  =  e  {1/  —u) 

Tell  what  the  above  formulas  mean. 

2  u  sin  a  " 
"  Horizontal  range  =  u  cos  a  X  * 


How  obtained  ? 

A  bullet  is  fired  with  a  velocity  of  i  000  feet  per 
second.  What  must  be  the  angle  of  inclination  in 
order  that  it  may  strike  a  point  in  the  same  horizontal 
plane  at  a  distance  of  15  625  feet  .-* 

Total  number  of  problems  taken  during  the  half-year,  about  180. 


1 5  2  ME  CHANICS-PKOBLEMS. 


STATICS 

HARVARD    UNIVERSITY 
First  Course  in  Mechanics 

1.  Find  the  components  of  a  force  of  500  pounds  along 
lines  inclined  to  it  by  (^)  0°  ;  {b)  24°;  {c)  30°.  Algebrai- 
cally only. 

2.  Find  the  moment  of  (300  pounds  48°  (-  4,  6)) 
about  {a)  (4,  6)  ;  {b)  (o,  o)  ;(.;(-  4,  6)  ;  {d)  (3,  -  7). 
Algebraically  only. 

3.  A  uniform  body  in  the  shape  of  an  isosceles  triangle 
with  base  of  60  feet  and  altitude  of  20  feet  weighs  200 
pounds.  It  is  supported  at  points  in  its  base  20  feet 
and  60  feet  respectively  from  the  left  end.  Forces  of  20 
pounds  and  40  pounds  act  vertically  upward  and  down- 
ward respectively  from  points  bisecting  the  left  and  right 
sloping  sides  respectively. 

Determine  the  pressures  upon  the  supports. 

4.  A  rectangle,  10  inches  by  8  inches,  has  one  corner 
at  the  origin,  two  sides  coincident  with  (9Jf  and  (9F,  and 
a  corner  at  (10,  8).  Two  forces,  of  20  pounds  each,  act 
one  along  the  upper  edge  of  it  toward  the  right,  the  other 
along  the  lower  edge  toward  the  left.  Two  more  forces, 
of  40  pounds  each,  act  respectively  upward  along  left 
edge  and  downward  along  the  right  edge. 

{a)    Is  the  body  subject  either  to  translation  or  rotation  ? 
(J?)    If  any  further  forces  be  needed  to  cause  equilibrium 
state  the  value  of  the  simplest  system  that  will  do  it. 


EXAMJNA  T/OA'S. 


I  S3 


5.  Find  the  center  of  gravity  of  a  plane  figure  of  five 
sides  with  corners  at  (o,  o),  (5,  o),  (4,  5),  (4,  3),  (14,  3), 
(8,  o). 

Solve  both  algebraically,  and  graphically,  using  in  the 
latter  case  the  general  string  polygon  method. 

6.  A  sphere  weighing  1000  pounds  rests  between  two 
smooth  planes  which  are  inclined  to  each  other  by  30°,  the 
less  steep  of  which  is  inclined  10°  to  the  horizontal. 

Determine  the  pressure  on  each  plane  algebraically. 

7.  A  plane  rectangular  frame  60  feet  high  and  10  feet 
wide  stands  on  two  supports,  one  at  each  of  the  lower 
corners.  A  horizontal  wind  force  of  4000  pounds  is 
applied  at  30  feet  from  the  ground  and  a  load  of  6000 
pounds  rests  at  the  middle  of  the  top. 

If  the  thrust  of  the  wind  be  assumed  to  be  resisted 
equally  by  the  supports,  determine  the  remaining  forces  at 
the  supports. 

8.  Determine  graphically  stresses  in  all  of  the  bars  of 
the  given  truss.  Show  numerical  results  upon  large  free- 
hand sketch  of  truss. 

9.  Determine  algebraically  stresses  in  Q,  7?,  and  S  of 
truss  of  last  question  without  finding  other  stresses. 

10.  Determine  the  reactions  {H^,  If.,  V^,  V.)  at  the 
supports  of  the  given  three-hinged  arch. 


a 

sin  a 

cos  a 

tan  a 

a 

Sin  a 

cos  a 

lana 

0 

0.00 

1. 00 

0.00 

48 

0.74 

0.67 

III 

10 

0.17 

0.88 

O.IO 

50 

0.77 

0.64 

1. 16 

24 

0.41 

0.91 

0.45 

60 

0.87 

0.50 

1.73 

30 

0.50 

0.89 

0.58 

80 

0.98 

0.17 

5.67 

40 

0.64 

0.77 

0.84 

45 

0.71 

0.71 

1. 00 

90 

I. CO 

0.00 

00 

184  MECHANICS-PROBLEMS. 


ANSWERS   TO    PROBLEMS. 

In  preparing  this  new  edition  two  opposing  sugges- 
tions have  been  offered  to  me :  one  that  I  should  give 
all  the  answers  to  the  problems,  the  other  that  I 
should  give  none.  I  have  taken  the  middle  ground, 
and  am  giving  about  half  of  the  answers,  believing 
that  this  method  will  serve  both  for  engineers  in  prac- 
tice and  others  who  wish  to  know  that  their  results 
are  correct,  and  for  college  classes  where  it  is  often 
preferred  that  some  of  the  answers  be  omitted  lest 
the  student  place  too  much  dependence  on  them.  It 
is  generally  agreed,  I  think,  that  with  students  the 
advice  frequently  given  by  Professor  Merriman  in  his 
excellent  text  books  should  be  emphasized,  namely : 
that  the  answers  are  not  the  main  part  of  a  problem. 
In  fact,  the  student  is  urged  not  to  consult  the  answer 
at  the  beginning  of  a  problem,  and  then  aim  merely 
to  get  that  numerical  result.  First  an  understanding 
of  the  problem  should  be  obtained,  then  a  diagram 
representing  the  data  should  be  drawn,  and  an  esti- 
mate of  the  answer  based  on  experience  should  be 
noted. 

Furthermore,  in  my  own  classes  I  require  that  the 
solutions  shall  be  carefully  made  in  special  note  books, 


ANSWERS   TO  PROBLEMS.  l8S 

and  that  the  student's  method  of  analysis  shall  be 
plain,  concise,  and  easily  understood ;  for  the  ability 
to  reason  soundly  and  to  demonstrate  clearly  should 
be  leading  aims  in  the  study  of  Mechanics. 

Work.  (3)  3  000  foot-pounds.  (6)  320  men.  (10) 
169  000  foot-pounds.  (ij.)  19  800  000  foot-pounds, 
(15)  352  000  foot-pounds.  (17)  104.8  foot-pounds.  (20) 
104  167  foot-pounds.  (22)  20  foot-pounds.  (25)  125 
pounds.  (28)  120  000  pounds.  (32)  1.5 1  inches.  (34) 
0.54  pounds.  (36)  28.5  pounds.  (38)  112  pounds. 
(40)  522.5  pounds.  (42)  6000  foot-pounds;  ratio  3  :  2. 
(44)  12.9  man-power.  (46)  9^\-  horse-power.  (48)  \\ 
horse-power.  (50)  36 j\  horse-power.  (53)  25  horse- 
power. (55)  435  horse-power.  (58)  139  kilowatts. 
(60)  67.8  amperes.  (63)  About  80  pounds  per  inch  of 
width.  (66)  4  inches.  (68)  4  horse-power.  (70)  12 
miles  an  hour.  (72)  15  pounds  per  ton.  (74)  i  000 
horse-power.  (77)  %%  against  friction;  23  520  000  foot- 
pounds wasted.  (79)  10.5  horse-power  theoretically. 
(81)  1 40  horse-power.  (82)  107  horse-power.  {^^  97.5 
horse-power.  (86)  i  665  horse-power.  (87)  13.2  horse- 
power. (89)  2  566  looms.  (91)  0.061.  (92)  12.6 
horse-power.  (95)  450  horse-power.  (97)  62  pounds 
per  square  inch.  (100)  5.6  horse-power.  (loi)  i  594 
horse-power.  (104)  9448100ms.  (106)  5  million  horse- 
power. (108)  0.39.  (no)  157  horse-power.  (113) 
132  no  000  foot-pounds,  or  132  million  Duty.  (116) 
10  500  cubic  feet.  (118)  3.44  machines.  (121)  21 
hours  18  minutes.  (124)  39  600  pounds.  (125)  0.14 
horse-power;  97  cubic  inches.  (127)  184' pounds.  (129) 
88  a  R/P  strokes  per  minute.  (131)  156  tons.  (133)  265 
pounds.       (137)   1 1-2     pounds    per    ton.       (140)   i  783 


1 86  MECHANICS-PROBLEMS, 

amperes.  (142)  393  pounds;  19.6  per  cent.  (145)  4.5 
feet.  (147)  I  856  horse-power.  (149)  12  758  foot- 
pounds; I  450  pounds.  (152)  0.17  horse-power.  (156) 
549  pounds.  (158)  I  250  tons.  (159)  15  625  feet 
height.  (162)  750  pounds.  (164)  44.3  turns.  (166) 
Ratio  I  to  1.94.  (167)  Average  of  28.75  foot-pounds. 
(169)  62.5  cubic  feet.  (171)  15.2  horse-power.  Force. 
(173)  300  pounds.  (176)  10  pounds.  (178)  43  units ; 
25.  (181)  29  pounds.  (183)  150  pounds;  90.  (186) 
Rafters  6.32  tons;  tie  rod  6  tons.  (188) '"Boom  77.3 
tons;  tackle  36.4.  (191)  50  pounds.  (193)  580  pounds. 
(194)  6  030  pounds.  (196)  117. 1  pounds;  82.8.  (199) 
2.8  pounds ;  9.6.  (200)  cos^^  =  bla,  b  being  distance 
from  C  to  AB  and  2  a  the  length  of  the  rod.  (203) 
Perpendicular  to  plane.  (205)  1020  pounds;  i  000. 
(209)  2.45  tons.  (211)  1. 15  tons.  (213)  86.6  pounds; 
100.  (217)  I  460  pounds;  i  990.  (218)  77  pounds. 
(220)  D  being  area  of  triangle,  Y  =  ^  b  {c^  -\-  c^  —  b^^  -r- 
4^D;  Q=W«(ft2^^2_^2y4^2?.  (222)  In  guy  10.3 
tons;  in  legs  18.6  tons.  (224)  Back  stay,  90  tons;  legs, 
100.  (226)  7.8  tons;  6.5.  (228)  Back  stays,  81  tons; 
A-frame  36;  wire  rope  29;  upper  boom  46;  lower  51; 
guy  or  tackle  49.5.  (229)  1.16  tons  ;  0.55  ;  0.53.  (232) 
14.2  pounds.  (233)  13  units  at  tan~^  j^  with  AB.  (234) 
2  V2^  parallel  to  CA  at  distance  from  AC  of  3  V2/2  X 
AB.  (238)  Ratio  i  to  V3.  (242)  Tension  =  W  / -^ 
2  \Ip  —  c^  smO.  (246)  115. 5  pounds;  57.7.  (252)  6250 
pounds;  5000,  (257)  6  inches  from  end.  (259)  9 
inches  from  middle;  18  pounds.  (263)  5  inches  from 
the  middle.  (266)  7  feet.  (268)  io6|  pounds.  (270) 
3.5  pounds;  ^  inch  from  middle.  (272)  4^  tons;  3§. 
(276)  6.5  pounds.      (278)  9.5  pounds.     (280)  11   inches 


ANSWERS    TO   PROBLEMS.  1 8/ 

and  6.  (282)  Posts  40  tons;  lower  chord,  37,5  and  32.5  ; 
upper  chord  32.5.  (284)  Post  6  tons;  tie  12.8;  chord 
12.5.  (286)  75  pounds.  (287)  33.6  pounds.  (290) 
540.  (292)  On  line  bisecting  vertical  angle  §  from  ver- 
tex. (293)  2  V3rt'/9,  '^a/T,,  4  V3  ^  from  the  sides,  if 
each  side  =  2  a.  (294)  6  V3^/ii,  3  ^f^a/ii,  2  V3  <?/ii 
from  sides;  outside  the  triangle  at  distance  (i^2>^/Sy 
—  3^3  ^/5»  2  V3  (i/s.  (296)  Any  point  of  line  parallel  to 
CD  passing  through  X  which  is  in  BC  produced  so  that 
CX  =  2  BC.  (299)  5  units  acting  parallel  to  BD,  cutting 
BC  produced  at  X,  so  that  4  CX  =  BC.  (302)  124 
pounds,  92,  134.  (306)  At  point  15  and  16  inches  from 
adjacent  sides.  (309)  2]  feet  from  rim.  (313)  If  D  be 
the  middle  point  of  BC,  R  is  represented  in  magnitude 
by  2  AD,  and  acts  through  X  parallel  to  DA,  X  being  in 
DC  orDB,  so  that  DX  =BC/8.  (315)  He  loses  i  pound. 
(317)  I  inch.  (318)  85.9  pounds.  (321)  15  pounds 
each.  (323)  At  C  force  is  horizontal,  and  =  W  V3/2  ; 
at  B  tan"^  V3/2  to  vertical  and  =  W  V7/2.  (326)  Length 
of  stick  from  nail  to  wall  v  3  :  pressure  :=  S^ ;^  ounces 
and  8  V -y^  _  I.  (327)  18900  pounds.  (328)  35.3 
feet.  (331)  P=  15  000  pounds.  (332)  |  of  length  from 
end  where  pressure  is  4  pounds.  (335)  1.33  inches. 
(339)  I  inch  from  AC,  i^  from  AB.  (342)  It  divides 
the  face  to  which  the  cover  is  hinged  in  ratio  of  i  to 
2.  (345)  From  left-hand  edge  2.84  inches;  5.36  inches. 
(348)  2  cos^  =  3  cos  (tt  —  ^)/3,  0  being  angle  with  hori- 
zontal. (350)  /i  =  r^^.  (352)  f  (355)  373  pounds. 
(357)  1/^/3-  (359)  200  pounds.  (362)  ^\.  (366)  I; 
incUnation,  tan~^  |.  (369)  433  pounds.  (371)  1140 
pounds;  314.  (374)  60°.  (376)  /a VVr/(i  4-  -y)  sin  a, 
r  being  radius,  and  W  the  weight  of  wheel.     (378)  47 


1 88  ME  CHANICS-PROBLEMS. 

feet.  (380)  1/V3.  (382)  104  pounds.  (388)  About 
2.08.  (393)  65  pounds.  (394)  2  800  pounds.  (396) 
898  pounds.  (398)  2>\'  (400)  13  inches.  (401)  37.6 
inches.  (405)  504  pounds.  (406)  1.92  horse-power. 
(407)  W/P  =  0.95  or  1.05.  (408)  137  thermal  units. 
(410)  0.44  horse-power.  (412)  3.5  horse-power.  Motion. 
(415)  35  feet.  (417)  13/^  miles  per  hour ;  27^-  (419) 
30  miles  an  hour.  (422)  150  feet;  200.  (424)  13 
pounds  per  ton.  (425)  99  feet.  (426)  5  seconds.  (428) 
6  seconds;  112  feet  per  second.  (431)  V<^^  feet 
per  second.  (433)  231  feet.  (435)  4080  feet.  (436) 
In  '^hl2  g  seconds,  and  3/^/4  feet  from  ground.  (438) 
350  feet.  (440)  7W3/2;  u/2.  (442)  17.6  feet.  (444) 
About  J  mile  up-stream.  (446)  245  miles  an  hour.  (449) 
Northwest  6  V2  miles  an  hour.  '(45^)  24  tt.  (454)  65.5 
miles  an  hour;  0.27  miles.  (457)  300  pounds.  (459)  24.3 
amperes  for  maximum  velocity.  (462)  1.9  miles.  (464) 
0.014.  (467)  16  V5  feet  per  second.  (469)  7.25  feet. 
(470)  8  feet  per  second.  (473)  3tV  pounds.  (475) 
6938  pounds.  (478)  {a)  ^^- g  feet;  {b)  70  pounds,  140 
and  i86f.  (479)  i  second.  (481)  15°  or  75°.  (483) 
44  V2  feet.  (486)  3903  feet  inside  of  city.  (487)  7.16 
miles.  (489)  8  600  feet ;  31  250  000  foot-pounds.  (494) 
0.3  tons.  (497}  2.83  pounds.  (500)  Tan-^  ^VVo-  (503) 
43  revolutions  per  minute.  (506)  5^  tons.  (507)  321 
tons.  (510)  23.ifeet.  (514)  3.4Pounds.  (515)  3666! 
feet.  (519)  3.1  feet  per  second.  (521)  496.8  feet  per 
second.  (525)  —  i  feet  per  second;  +2.  (527)  A 
returns  5  feet  per  second  ;  B  moves  at  45°  with  its  course 
and  velocity  of  10  V2.  Review.  (529)  (a)  58  tons- 
(531)  56.7  per  cent.  (534)  60  000  pounds  close  to  tower ; 
47000   in   middle.      (535)  52222    pounds.      (536)   17 


ANSWERS    TO  PROBLEMS.  1 89 

revolutions.  (538)  7.25  feet.  (541)  In  bolt  27500 
pounds.  (543)  In  leg  63.24  tons ;  in  inclined  members, 
18.75  tons.  (545)  0.036.  (549)  I  250  pounds.  (552) 
27.8  feet  per  second.  (557)  472  tons.  (558)  160. 
(559)  682  revolutions  per  minute;  1910  amperes  for  the 
8  motors.  (560)  i  453  horse-power.  (563)  0.19.  (564) 
16  879  pounds;  8.1  per  cent.  (566)  2  720  pounds  per 
square  inch.  (568)  cos  d  =  P  2  W.  (571)  Horizontal 
stress  1.8  tons;  vertical  additional  0.9  tons.  (573)  2.0 
horse-power.  (574)  35  pounds.  (575)  {a)  41  800  pounds  ; 
{h)  458000  pounds;  {c)  i  195  gallons.  (577)  7419 
pounds.  (578)  30  862  pounds.  (579)  25  800  pounds, 
35  900.  (580)  I  780  pounds.  (583)  In  air,  boom  16.57 
tons;  in  water,  boom  14.45  tons.  (5S6)  4.V  tons.  (587) 
15.2  feet.  (589)  33.75  feet;  ^V  second.  (590)  9  518 
pounds.  (592)  {b)  19.2  inches.  (594)  54.2  pounds; 
10.4.     (596)  80,6  per  cent,     (600)  42  horse-power. 


190 


ME  CII4  NICS-PR  OBLEMS. 


Falling  Bodies :  Velocity  Acquired  by  a  Body  Falling  a 
Olveu  Helgbt. 


s 

i 

A 

^ 

S 

>y 

«• 

^ 

*3 

>^ 

^ 

.C 

^ 

^ 

1 

.6X1 

'5 

0 

^ 

1 

[m 

1 

•^ 

'0 
0 

:! 

% 

<D 

.-H 

'53 

<a 

'S 

0 

0) 

» 

^ 

n 

> 

K 

I 

m 

"3 
> 

rt 

I 

w 

I 

feet. 

feet 

feet. 

feet 

feet. 

feet 

feet. 

feet 

feet. 

feet 

feet. 

feet 

p.  see. 

p. sec. 

p. sec. 

p.  sec. 

p  sec. 

p.  sec. 

.005 

.57 

.39 

5.01 

1.20 

8.79 

5. 

17.9 

23. 

38.5 

72 

68.1 

,010 

.80 

.40 

5.07 

1.22 

8.87 

.2 

18.3 

.5 

38.9 

73 

68.5 

.015 

.98 

.41 

5.14 

1.24 

8.94 

.4 

18.7 

24. 

39.3 

74 

69.0 

.020 

1.13 

.42 

5.20 

1.26 

9.01 

.6 

19.0 

.5 

39.7 

75 

69.5 

.025 

1.27 

.43 

5.26 

1.28 

9.08 

.8 

19.3 

25 

40.1 

76 

69.9 

.030 

1.39 

.44 

5.32 

1.30 

9.15 

6. 

19.7 

26 

40.9 

77 

70.4 

.035 

1.50 

.45 

5.38 

1.32 

9.21 

.2 

20.0 

27 

41.7 

78 

70.9 

.040 

1.60 

.46 

5.44 

1.34 

9.29 

.4 

20.3 

28 

42.5 

79 

71.3 

.045 

1.70 

.47 

5.50 

1.36 

9.36 

.6 

20  6 

29 

43.2 

80 

71.8 

.050 

1.79 

.48 

5.56 

1.38 

9.43 

.8 

20.9 

30 

43.9 

81 

72.2 

.055 

1.88 

.49 

5.61 

1.40 

9.49 

7. 

21.2 

31 

44.7 

82 

72.6 

.060 

1.97 

.50 

5.67 

1.42 

9.57 

.2 

21.5 

82 

45.4 

83 

73.1 

.065 

2.04 

.51 

5.73 

1.44 

9.62 

.4 

21.8 

33 

46.1 

84 

73.5 

.070 

2.12 

.52 

5.78 

1.46 

9.70 

.6 

22.1 

84 

46.8 

85 

74.0 

.075 

2.20 

.53 

5.84 

1.48 

9.77 

.8 

22.4 

85 

47.4 

86 

74.4 

.080 

2.27 

.54 

5.90 

1.50 

9.82 

8. 

22.7 

36 

48.1 

87 

74.8 

.085 

2.34 

.55 

5.95 

1.52 

9.90 

.2 

23.0 

37 

48.8 

88 

75.3 

.090 

2.41 

.56 

6.00 

1.54 

9.96 

.4 

23.3 

88 

49.4 

89 

75.7 

.095 

2.47 

.57 

6.06 

1.56 

10.0 

.6 

23.5 

89 

50.1 

90 

76.1 

.100 

2.54 

.58 

6.11 

1.58 

10.1 

.8 

23.8 

40 

50.7 

01 

76.5 

.105 

2.60 

.59 

6.16 

1.60 

10.2 

9. 

24.1 

41 

51.4 

92 

76.9 

.110 

2.66 

.60 

6.21 

1.65 

10.3 

.2 

24.3 

42 

52.0 

93 

77.4 

,115 

2.72 

.62 

6.32 

1.70 

10.5 

.4 

24.6 

43 

52.6 

94 

77.8 

.120 

2.78 

.64 

6.42 

1.75 

10.6 

.6 

24.8 

44 

53.2 

95 

78.2 

.125 

2.84 

.66 

6.52 

1.80 

10.8 

.8 

25.1 

45 

53.8 

96 

78.6 

.130 

2.89 

.68 

6.61 

1.90 

11.1 

10. 

25.4 

46 

54.4 

97 

79.0 

.34 

3.00 

.70 

6.71 

2. 

11.4 

.5 

26.0 

47 

65.0 

98 

79.4 

.15 

3.11 

.72 

6.81 

2.1 

11.7 

11. 

26.6 

48 

55.6 

99 

79.8 

.10 

3.21 

.74 

6.90 

2.2 

11.9 

.5 

27.2 

49 

56.1 

100 

80.2 

.17 

3.31 

.76 

6.99 

2.3 

12.2 

12. 

27.8 

50 

56.7 

125 

89.7 

.18 

3.40 

.78 

7.09 

2.4 

12.4 

.5 

28.4 

51 

57.8 

150 

98.3 

.19 

8.50 

.80 

7.18 

2.5 

12.6 

13. 

28.9 

52 

57.8 

175 

106 

.20 

3.59 

.82 

7.26 

2.6 

12. C 

.5 

29.5 

53 

58.4 

200 

114 

.21 

3.68 

.84 

7.35 

2.7 

13.2 

14. 

30.0 

54 

59.0 

225 

120 

.22 

8.76 

.86 

7.44 

2.8 

13.4 

.5 

3  .5 

55 

59.5 

250 

126 

.23 

3.85 

.88 

7.53 

2.9 

13.7 

15. 

31.1 

56 

60.0 

275 

133 

.24 

8.93 

.90 

7.61 

3. 

139 

.5 

31.6 

57 

60.6 

300 

139 

.25 

4.01 

.92 

7.69 

3.1 

14.1 

16. 

32.1 

58 

61.1 

350 

150 

.26 

4.09 

,94 

7.78 

3.2 

14.3 

.5 

32.6 

59 

61.6 

400 

160 

.27 

4.17 

.96 

7.86 

3.3 

14.5 

17. 

S3.1 

60 

62.1 

450 

170 

.28 

4  25 

.98 

7.94 

3.4 

14.8 

.5 

33.6 

61 

62  7 

600 

179 

.29 

4.32 

1.00 

8.02 

3.5 

15.0 

18. 

34.0 

"'i 

60.2 

550 

188 

.30 

4.39 

1.02 

8.10 

3.6 

15.2 

.5 

34.5 

63 

63.7 

600 

197 

.31 

4.47 

1.04 

8.18 

3.7 

15.4 

19. 

35.0 

64 

64.2 

700 

212 

.32 

4.54 

1.06 

8.26 

3.8 

15.6 

.5 

35.4 

65 

64.7 

800 

227 

.33 

4.61 

1.08 

8.34 

3.9 

15.8 

20. 

35.9 

66 

65.2 

900 

241 

.34 

4.68 

1.10 

8.41 

4. 

16.0 

.5 

36.3 

67 

65.7 

1000 

254 

.85 

4.74 

1.12 

8.49 

.2 

16.4 

21. 

36.8 

68 

66.1 

2000 

359 

.36 

4.81 

1.14 

8.57 

.4 

16.8 

.5 

37.2 

69 

66.6 

3000 

439 

.37 

4.88 

1.16 

8.64 

.6 

17.2 

22. 

37.6 

70 

67.1 

4000 

507 

.38 

4.94 

1.18 

8.72 

.8 

17.6 

.5 

38.1 

71 

67.6 

5000 

567 

Reprinted  from  Kent's  Mechanical  Engineers'  Pocket-Book. 


Functions  of  Angles 


Angle 

Sin 

Tan 

Sec 

Cosec 

Cot 

cos   1     1 

O 

0. 

0. 

.0 

00 

00 

I. 

90 

I 

0.0175 

0.0175 

I  000 1 

57^299 

57.290 

0.9998 

89 

2 

.0349 

•0349 

1.0006 

28654 

28.636 

•9994 

88 

3 

•0523 

.0524 

I. 0014 

19.107 

19.081 

.9986 

87 

4 

.0698 

.0699 

1.0024 

14336 

14.301 

.9976 

86 

5 

.0872 

.0875 

1 .0038 

11.474 

1 1 .430 

.9962 

85 

6 

0.1045 

O.I  05 1 

1.0055 

9.5668 

9-5M4 

0.9945 

84 

7 

.1219 

.1228 

1.0075 

8.2055 

8.1443 

•9925 

83 

8 

.1392 

.1405 

1 .0098 

7-1853 

7-1154 

■9903 

82 

9 

.1564 

.«584 

1. 0125 

63925 

6.3138 

-9877 

81 

10 

.'736 

.1763 

I.OI54 

5-7588 

5-6713 

.9848 

80 

II 

O.IQ08 

0.1944 

I.O187 

52408 

5.1446 

0.9816 

79 

12 

.2079 

.2126 

1.0223 

4.8097 

4.7046 

.9781 

78 

13 

.2250 

.2309 

1.0263 

4  4454 

4-33'5 

.9744 

77 

14 

.2419 

•2493 

1 .0306 

4-1336 

4.0108 

•9703 

76 

15 

.2588 

.2679 

1-0353 

3-8637 

37321 

-9659 

75 

i6 

0.2756 

0.2867 

1.0403 

3.6280 

3-4874 

0.9613 

74 

^7 

.2924 

•3057 

1.0457 

34203 

3.2709 

•9563 

73 

i8 

.3090 

.3249 

I.0515 

32361 

30777 

.9511 

72 

19 

•3256 

•3443 

1.0576 

3.0716 

2.9042 

•9455 

71 

20 

.3420 

.3640 

1.0642 

2.9238 

2-7475 

•9397 

70 

21 

0.3584 

03839 

1. 07  I  2 

2.7904 

2.6051 

0.9336 

69 

22 

•3746 

.4040 

1.0785 

2.6695 

2-4751 

.9272 

68 

23 

•3907 

.4245 

1 .0864 

2-5593 

2.^3559 

•9205 

67 

24 

.4067 

.4452 

1 .0946 

2.4586 

2.2460 

•9135 

66 

25 

.4226 

.4663 

I. 1034 

2.3662 

2-1445 

.9063 

65 

26 

0.4384 

0.4877 

I.II26 

2.2812 

2-0503 

0.8988 

64 

27 

.4540 

•5095 

1. 1223 

2.2027 

1.9626 

,8910 

63 

28 

.4695 

•5317 

1. 1326 

2.1301 

1.8807 

.8829 

62 

29 

.4848 

•5543 

i^i434 

2.0627 

1 .8040 

.8746 

61 

30 

.5000 

•5774 

1-1547 

2.0000 

1.7321 

.8660 

60 

31 

0.5150 

0.6009 

1. 1666 

1. 94 16 

1.6643 

0.8572 

59 

32 

•5299 

.6249 

1. 1792 

1. 887 1 

1.6003 

.8480 

58 

33 

•5446 

.6494 

1. 1924 

I. 8361 

1-5309 

.8387 

57 

34 

•5592 

•6745 

1.2062 

1.7883 

1.4826 

.8290 

56 

35 

■573^ 

.7002 

1.2208 

1-7435 

1.4281 

.8192 

55 

36 

0.5878 

0.7265 

1.2361 

1.7013 

1-3764 

0.8090 

54 

37 

.6018 

•7536 

1-2521 

1. 6616 

1.3270 

.7986 

53 

38 

.6157 

•7813 

1.2690 

1.6243 

1.2799 

.7880 

52 

39 

.6293 

.8098 

1.2868 

1.5890 

1-2349 

•7771 

51 

40 

.6428 

.8391 

13054 

1-5557 

1. 1918 

.7660 

50 

41 

0.6561 

0.8693 

1.3250 

1-5243 

1.1504 

0.7547 

49 

42 

.6691 

.9004 

1.3456 

1-4945 

1.1106 

•7431 

48 

43 

.6820 

•9325 

1.3673 

1.4663 

1.0724 

■73H 

47 

44 

.6947 

.9657 

1.3902 

1.4396 

10355 

•7193 

46 

45 

.7071 

1.4142 

1.4142 

I. 

.7071 

45 

[_ 

Cos 

Cot 

Cosec 

Sec 

Tan 

Sin 

Angle 

A    FEW   IMPORTANT    UNIT    VALUES 
BE    USED   IN    SOLVING  THESE 
PROBLEMS 


TO 


30  miles 

an  hour 

=:  44  feet  per  second 

I  ton 

•=  2  000  pounds 

I  fathom 

=  6  feet 

I  knot 

=  6  080  feet 

I  cubic  foot  of  water 

=  62^  pounds 

=  7^  gallons 

I  gallon 

of  water 

=  8^  pounds 

I  pound 

of  watei 

:  pressure 

=  2.304  feet  head 

I  British  thermal 

1  unit 

=  778  foot-pounds  of  energy 

g,  acceleration  ol 

■  gravity 

=  32  feet  per  second  per 

second, 

unless  otherwise  specified 

e  the  base  of  Napierian  system  of 

logarithms 

=  2.7  1828  1828 

I  horse-] 

power 

=  746  watts 

I  kilowatt 

=  1.34  horse-power 

Watts  =  volts  X  amperes 

IMPORTANT 

FUNCTIONS    OF    ANGLES  AND    TRIGO- 

NOMETRIC   RELATIONS. 

0° 

30° 

45° 

60°               90 

120° 

I 

I 

Vi 

V3 

2 

V2 

2 

2 

Sine 

0 

.500 

.707 

.866 

.866 

^Z 

I 

I 

—                0 

_i 

2 

V2 

2                 ° 

2 

Cosine 

I 

.866 

.707 

.500 

-.500 

Vi 

I 

y/^           Infinite 

-V3 

Tangent 

0 

.577 

1.732 

-  1.732 

Sin  = 

Perp 
Hypot 

Cos  = 

Base                   ^^^       Perp 
Hypot                               Base 

Tan  = 

Sin 
Cos 

Cot  = 

Sec- ,.' 
Tan                                  Cos 

Cosec  = 

I 

Vers  = 

I  —  cos                a\b  =>  Sin  A 

:Sin^ 

'Sin 

•  B)^^inACosB-\-Cc 

Sin  {A  4 

)sASinB     c^Va^  +  l>^-2abCosC 

192 

INDEX, 


93 


INDEX 


Acceleration,  5,  119 
Angular  velocity,  128 
Answers  to  problems, 
Automobile,  22,  140 
Axle  friction,  115 


:84 


Belt  friction,  108,  1 70 
Bicycles,  140 
Bolt  Friction,  104,  150 
Bridges,  75,  77.  78, 147,  154 

Cast-iron  pipe,  166 
Center  of  gravity,  4,  90 
Centrifugal  force,  5,  137 
Centroid,  4 
Chimney,  145 
Coal,  unloading,  23,  153 

wagon,  90 
Coefficient  of  friction,  96 
Components,  3,  4,  98,  100 
Concurrent  forces,  3 
Cooper's  loading,  157 
Couples,  4,  84 

Dam  falling,  149 
Davit,  77 
Definitions,  2 
Derricks,  54,  167 
Dipper  dredge,  64,  65 
Drum  for  hoisting,  169 

Electric  car,  131 

current,  19 
motors,  41,  129 

Energy,  1,44 


Equilibriant,  3 
Examination  papers,  174 

Flailing  bodies,  123 
Fire  engines,  37 
streams,  49 
Floor-posts,  83,  102 
Floating  cantilever,  151 
Fly  wheels,  48,  140,  148 
Foot-pounds,  7 
Foot-step  bearings,  117 
Force  problems,  51 
Forces  at  a  point,  5 1 
Fortification  wall,  135 
Friction  coefficients,  96 

problems,  96 
Friction  of  angles,  191 

Gas  engines,  16,  171 
Governors,  139,  172 
Gravity,  acceleration  of,  44,  123, 
190 

Horse-power,  2,  16 

Impulse,  5,  142 
Indicator  cards,  26 

Ladders,  79,  103,  104 

Launching  data,  45,  151 

Least  pull,  100 

Levers,  73 

Locomotives,  22,41,122, 128, 130, 

141,  156 
Logarithmic-decimal  paper,  108 


194 


MECJlANJCS-PROnLEMS. 


Moments,  3,  72 
Momentum,  142 
Motion  problems,  119 

Parallel  forces,  72,  80 
Parallelogram  of  forces,  3 
Pendulum,  141 
Pile  driver,  12 

resistance,  145 
Plates,  structural,  91 
Projectiles,  46,  134 
Pulleys,  13,21 
Pumps,  35,  38 

Rail  sections,  92 
Relative  velocity,  126 
Resolution  of  forces,  100 
Restitution,  coefficient,  5 
Resultant,  3 
Retaining  walls,  89,  167 
Review  problem,  145 
Roof  trusses,  70 
Rope  friction,  106,  114,  173 


Sailing  vessel,  68 

Shears,  61,  62,  63 

Ship  resistance,  40,  151 

Sound  velocity,  123 

Steam  engines,  25,  56,  89,  171 

turbine  shaft,  1 18 
Steel  rails,  93 
Structural  plates,  92 

Trench  machine,  160 

Tripod,  66 

Trusses,  70,  71,  72,  77,  78 

Unit  values,  192 

Velocities,  119 

of  falling  bodies,  190 
Water  gates,  97,  164,  168 

motor,  20,  162 

turbine,  ^iZi  14^ 

power,  i,T,,  38 
Water-works  tanks,  163 
Wedges,  loi 
Wire  rope,  173 
Work  problems,  7 


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